OpenMolcas: From Source Code to Insight

OpenMolcas: From Source Code to InsightIgnacio Fdez. Galvan,1,2 Morgane Vacher,1 Ali Alavi,3 Celestino Angeli,4 Francesco Aquilante,5

Jochen Autschbach,6 Jie J. Bao,7 Sergey I. Bokarev,8 Nikolay A. Bogdanov,3 Rebecca K. Carlson,7,9

Liviu F. Chibotaru,10 Joel Creutzberg,11,12 Nike Dattani,13 Mickael G. Delcey,1 Sijia S. Dong,7,14

Andreas Dreuw,15 Leon Freitag,16 Luis Manuel Frutos,17 Laura Gagliardi,7 Frederic Gendron,6

Angelo Giussani,18,19 Leticia Gonzalez,20 Gilbert Grell,8 Meiyuan Guo,1,21 Chad E. Hoyer,7,22

Marcus Johansson,12 Sebastian Keller,16 Stefan Knecht,16 Goran Kovacevic,23 Erik Kallman,1

Giovanni Li Manni,3 Marcus Lundberg,1 Yingjin Ma,16 Sebastian Mai,20 Joao Pedro Malhado,24

Per Åke Malmqvist,12 Philipp Marquetand,20 Stefanie A. Mewes,15,25 Jesper Norell,11

Massimo Olivucci,26,27,28 Markus Oppel,20 Quan Manh Phung,29 Kristine Pierloot,29 Felix Plasser,30

Markus Reiher,16 Andrew M. Sand,7,31 Igor Schapiro,32 Prachi Sharma,7 Christopher J. Stein,16,33

Lasse Kragh Sørensen,1,34 Donald G. Truhlar,7 Mihkel Ugandi,1,35 Liviu Ungur,36

Alessio Valentini,37 Steven Vancoillie,12 Valera Veryazov,12 Oskar Weser,3 Tomasz A. Wesołowski,5

Per-Olof Widmark,12 Sebastian Wouters,38 Alexander Zech,5 J. Patrick Zobel,12

and Roland Lindh*,2,39

1Department of Chemistry − Ångstrom Laboratory, Uppsala University, P.O. Box 538, SE-751 21 Uppsala, Sweden2Department of Chemistry − BMC, Uppsala University, P.O. Box 576, SE-751 23 Uppsala, Sweden3Max Planck Institut fur Festkorperforschung, Heisenbergstraße 1, 70569 Stuttgart, Germany4Dipartimento di Scienze Chimiche e Farmaceutiche, Universita di Ferrara, Via Luigi Borsari 46, 44121 Ferrara, Italy5Departement de Chimie Physique, Universite de Geneve, 30 quai Ernest-Ansermet, CH-1211 Geneve 4, Switzerland6Department of Chemistry, University at Buffalo, State University of New York, Buffalo, New York 14260-3000, United States7Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota,Minneapolis, Minnesota 55455-0431, United States8Institut fur Physik, Universitat Rostock, Albert-Einstein-Straße 23-24, 18059 Rostock, Germany10Theory of Nanomaterials Group, University of Leuven, Celestijnenlaan 200F, Leuven 3001, Belgium11Department of Physics, AlbaNova University Center, Stockholm University, SE-106 91 Stockholm, Sweden12Division of Theoretical Chemistry, Kemicentrum, Lund University, P.O. Box 124, SE-221 00 Lund, Sweden13Harvard Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, United States15Interdisciplinary Center for Scientific Computing, Heidelberg University, Im Neuenheimer Feld 205 A, 69120 Heidelberg,Germany16Laboratorium fur Physikalische Chemie, ETH Zurich, Vladimir-Prelog-Weg 2, 8093 Zurich, Switzerland17Departamento de Química Analítica, Química Física e Ingeniería Química, and Instituto de Investigacion Química “Andres M. delRío”, Universidad de Alcala, E-28871 Alcala de Henares, Madrid, Spain18Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, United Kingdom19Instituto de Ciencia Molecular, Universitat de Valencia, Apartado 22085, ES-46071 Valencia, Spain20Institute of Theoretical Chemistry, Faculty of Chemistry, University of Vienna, Wahringer Straße 17, 1090 Vienna, Austria23Division of Materials Physics, Ruđer Boskovic Institute, P.O.B. 180, Bijenicka 54, HR-10002 Zagreb, Croatia24Department of Chemistry, Imperial College London, London SW7 2AZ, United Kingdom25Centre for Theoretical Chemistry and Physics, The New Zealand Institute for Advanced Study (NZIAS), Massey UniversityAlbany, Private Bag 102904, Auckland 0632, New Zealand26Department of Biotechnology, Chemistry and Pharmacy, University of Siena, via A. Moro 2, 53100 Siena, Italy27Department of Chemistry, Bowling Green State University, Bowling Green, Ohio 43403, United States28USIAS and Institut de Physique et Chimie des Materiaux de Strasbourg, Universite de Strasbourg-CNRS, 67034 Strasbourg,France

Received: May 31, 2019Published: September 11, 2019

Article

pubs.acs.org/JCTCCite This: J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

© XXXX American Chemical Society A DOI: 10.1021/acs.jctc.9b00532J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

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http://dx.doi.org/10.1021/acs.jctc.9b00532

29Department of Chemistry, KU Leuven, Celestijnenlaan 200F, Leuven 3001, Belgium30Department of Chemistry, Loughborough University, Loughborough LE11 3TU, United Kingdom32Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem, Israel36Department of Chemistry, National University of Singapore, 117543 Singapore37Theoretical Physical Chemistry, Research Unit MolSys, Allee du 6 Aout, 11, 4000 Liege, Belgium38Brantsandpatents, Pauline van Pottelsberghelaan 24, 9051 Sint-Denijs-Westrem, Belgium39Uppsala Center for Computational Chemistry (UC3), Uppsala University, P.O. Box 596, SE-751 24 Uppsala, Sweden

*S Supporting Information

ABSTRACT: In this Article we describe the OpenMolcasenvironment and invite the computational chemistry commun-ity to collaborate. The open-source project already includes alarge number of new developments realized during thetransition from the commercial MOLCAS product to theopen-source platform. The paper initially describes thetechnical details of the new software development platform.This is followed by brief presentations of many new methods,implementations, and features of the OpenMolcas programsuite. These developments include novel wave functionmethods such as stochastic complete active space self-consistent field, density matrix renormalization group(DMRG) methods, and hybrid multiconfigurational wavefunction and density functional theory models. Some of these implementations include an array of additional options andfunctionalities. The paper proceeds and describes developments related to explorations of potential energy surfaces. Here wepresent methods for the optimization of conical intersections, the simulation of adiabatic and nonadiabatic molecular dynamics,and interfaces to tools for semiclassical and quantum mechanical nuclear dynamics. Furthermore, the Article describes featuresunique to simulations of spectroscopic and magnetic phenomena such as the exact semiclassical description of the interactionbetween light and matter, various X-ray processes, magnetic circular dichroism, and properties. Finally, the paper describes anumber of built-in and add-on features to support the OpenMolcas platform with postcalculation analysis and visualization, amultiscale simulation option using frozen-density embedding theory, and new electronic and muonic basis sets.

1. INTRODUCTIONAt the 4th MOLCAS developers’ workshop, 30 March−1 April2016, in Vienna, Austria, the MOLCAS developer communitydecided that the source code of the MOLCAS project wouldbe released, where the authors agree, as an open-source projectunder the GNU Lesser General Public License (LGPL)1 − theOpenMolcas project. This decision was followed by intensivework to adapt to this new format and context. An open-sourcerepository was established September 2017. In late 2018 thetransition to the new open-source format and integrationplatform was completed. The history of the OpenMolcascodebase dates back much earlier in time, and for a summaryof it we refer to previous MOLCAS publications.2−8 Here wereport on the new and recent developments and optionsavailable in OpenMolcas.OpenMolcas is a software package capable of performing

state-of-the-art quantum chemical calculations, but it is not theonly one. Other packages, some of them open source, withsimilar or overlapping capabilities are (a necessarily non-exhaustive list) as follows: ACES,9 ADF,10 BAGEL,11

BigDFT,12 CFOUR,13 Columbus,14 Dalton,15 deMon2k,16

DIRAC,17 Firefly,18 GAMESS,19 GAMESS-UK,20 Gaussian,21

HONDOPLUS,22 Jaguar,23 Molpro,24 MRCC,25 NWChem,26

NRLMOL,27 ORCA,28 PQS,29 Psi4,30 PySCF,31 Q-Chem,32

TeraChem,33 Turbomole.34 A more comprehensive relation,but restricted to open-source programs, can be found in ref 35.A detailed comparison of the OpenMolcas features with theseother packages is out of the scope of this report, but it can be

stated that the main strength and focus of OpenMolcas is onmulticonfigurational wave function methods, and applicationsand properties for which these methods are appropriatelysuited, as reflected in the contents of this article.Apart from scientific and methodological improvements, the

release of OpenMolcas incorporates a series of changes in codemanagement and tools. These changes, as presented here, areaimed at an optimal interface for software accessibility for newdevelopers and facilitating interaction with other codes. In thisreport the new developments are subdivided into five differentcategories: novel wave function and density functionalmethods, approaches to explore potential energy surfaces(PESs), implementations associated with various types ofspectroscopy, tools for postanalysis of orbitals and wavefunctions, and finally a set of some miscellaneous develop-ments.On the issue of resolving the prohibitive exponential scaling

of the multiconfigurational wave function a number of newtechniques have been introduced to eliminate this bottleneck.In particular, the Stochastic-CASSCF method has beenimplemented. Furthermore, two interfaces are introduced−with the CheMPS2 and QCMAQUIS programs−that greatlyexpand the potential of multiconfigurational wave functions byallowing much larger active spaces as well as speeding upcalculations with usual active spaces. This is achieved bysupplementing the full configuration interaction (full CI, FCI)solver in the RASSCF module by the density matrixrenormalization group (DMRG). With these interfaces it is

Journal of Chemical Theory and Computation Article

DOI: 10.1021/acs.jctc.9b00532J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

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possible to perform DMRG-SCF, DMRG-CASPT2, andDMRG-NEVPT2 calculations, obtain analytical DMRG-SCFgradients, or compute state interaction. Moreover, themulticonfiguration pair-density functional theory (MC-PDFT) option has been further developed. This includestranslating several KS-DFT functionals not previously avail-able, introducing analytical gradients for state-specific CAS-PDFT, state-interaction PDFT (SI-PDFT) for situations wheretwo electronic states are strongly interacting and the use ofPDFT in association with DMRG technology.The exploration of the PESs is a significant part of any

sophisticated tool for electronic structure calculations. In spiteof being a leading package for the computational study ofelectronic excited states, MOLCAS has for a long time lackedthe ability of computing nonadiabatic coupling vectors. Therecent implementation of analytical nonadiabatic couplingvectors for state-averaged CASSCF wave functions is nowavailable, together with practical methods for optimizing andcharacterizing conical intersections, and the use in simulationsof nonadiabatic processes. Furthermore, simulating the timeevolution of matter in the vicinity of electronic degeneracyregions is essential for understanding the rate and the productdistribution of photochemical reactions and for complement-ing time-resolved experiments. Direct dynamics methods weredeveloped to avoid the required computation and fitting of thePESs prior to dynamics simulations and to address the issue ofthe exponential scaling with the system size. OpenMolcas nowsupports three different tools for this. First, the DYNAMIX andSURFACEHOP modules of OpenMolcas allow moleculardynamics (MD) simulations of adiabatic and nonadiabaticprocesses; there are now options for generating startingconditions, adding external forces to the molecular system forsteered MD simulations, and simulating isotope effects.Second, the SHARC−OpenMolcas interface allows thesimulation of nonadiabatic trajectory surface hopping dynam-ics including any arbitrary coupling; this means it can deal withphotochemical events involving internal conversion andintersystem crossing on the same footing. Third, we reporthere a new interface to the Quantics package. Dynamics codefor classical nuclei is rather straightforward since information isonly needed at a given position in space; a quantumdescription of the nuclei has more problems because ofdelocalization of the wavepacket. The new interface with theQuantics code allows a quantum mechanical treatment of bothelectronic and nuclear dynamics via the direct-dynamicsvariational multiconfiguration Gaussian (DD-vMCG) method.The OpenMolcas package can generate operator matrix

elements between spin−orbit-coupled electronic states and isthus frequently used in simulation of various spectroscopicprocesses. Recently, these tools have been improved. For high-energy photons the short wavelength means that the electricfield can no longer be treated as constant−as in the dipoleapproximation−and electric-dipole forbidden transitions canbe observed. The solution, as presented here, is to eitherintroduce higher orders in the multipole expansion or, moreelegantly, use the exact exponential form of the wave vector.Core-hole states are notoriously problematic to handle instandard computations, since these states are high up inenergy. Now there is a simple option to use projection duringthe CI optimization to prevent the core-hole being filled.Moreover, spectroscopy typically involves transitions betweenan initial state and several final states. The standard CIalgorithms are designed to calculate ground and a relatively

small number of low-lying excited states. However, to simulateexperiments as X-ray spectroscopy, hundreds of high-energystates are needed. A new approach to the CI problem, whichefficiently handles this case, is presented here. We report alsothe implementation of multiconfigurational Dyson orbitals foraccurate simulations of molecular ionization processes.Another example is magnetic circular dichroism (MCD) asone of the prime tools to study the electronic states ofcomplicated metal complexes and metalloproteins. Softwaretools have now been created for the easy generation of MCDspectral intensities with OpenMolcas. We will also discuss thecomputation of magnetic properties for transition metal andlanthanide compounds. Moreover, an example in which theOpenMolcas platform has been used as an alternative toexperiments, to generate microhartree accuracy of ionizationpotentials, is given.Any quantum chemical simulation needs the support of

postcalculation analysis tools to render the simulations to bemore than just the matter of computing a numerical value ofthe energy or some other molecular property. Such toolsenable the scientist to make qualitative conclusions from thesimulations and to obtain insight in addition to quantitativepredictions. They typically involve the generation of orbitals,which could be natural, canonical, or localized, and thegraphical representation of these. In that respect, OpenMolcashas recently been enhanced with the following new utilities.The graphical user interface LUSCUS is a next generation ofGV (Graphical Viewer) code; it is a lightweight viewer of inputand output data produced by various quantum chemical codes,in particular the OpenMolcas program suite. One of thebottlenecks in the graphical visualization procedure is the sizeof the intermediate data. The ability to on-demand computethe data for visualization is now implemented in a stand-alonecode, SAGIT. Furthermore, Pegamoid is a new orbital viewerthat can read and write the native OpenMolcas orbital formats,simplifying the process of selecting and analyzing the activespace for MCSCF calculations. Moreover, binatural orbitals,also called natural transition orbitals, are now available in orderto characterize the nature of an electronic excitation. This canbe computed by singular value decomposition of a transitiondensity matrix using RASSCF wave functions in the RASSImodule. Finally, the WFA (wave function analysis) module notonly provides visualization tools but also computes quantitativedescriptors, which allow for a rigorous and completelyautomatized analysis of excited-state computations.In addition, as listed below, miscellaneous utilities and

functionalities have been added to the program suite. Therelativistic ANO-RCC basis set has been the spearhead of theMOLCAS/OpenMolcas program since its introduction adecade ago. It is now accompanied by an extremely smallrelativistic alternative, the ANO-XS basis set. The small size ofANO-XS greatly reduces integral computation times yet stillpredicts, e.g., geometries or excitation energies with only smalldifferences as compared to the larger ANO-RCC basis sets.Moreover, OpenMolcas is adapted to use point groups beyondD2h. Thus, for highly symmetrical molecules OpenMolcas canoccasionally generate wave functions with broken symmetry. Itis a problematic programming task to adapt a computationalcode to use higher point groups. However, a code for aposteriori symmetrization of the wave function, msym, hasbeen developed and implemented in OpenMolcas. As a furtherdevelopment we report on the ability to simulate muonicatoms and molecules, i.e., systems containing one muon−a



C


Fermion with a mass about 207 times heavier than an electron.This includes developments of muonic basis sets, bookkeepingof two types of Fermionic particles and basis sets in integraland wave function code, diagonal finite mass correction, andanalysis of parameters associated with the finite-nucleusapproximation. Finally, a new method for multiscale electronicstructure calculations has been implemented which is based onfrozen-density embedding theory (FDET).Thus, below we will in some detail describe and discuss

these new tools in the OpenMolcas program suite. Thestructure of the Article will in all respects follow the order aspresented above. This will be followed with a brief summary.

2. CODE DEVELOPMENT AND TOOLSThe tools and culture in software development have changeddramatically in the last three decades since the initial versionsof MOLCAS. The original code “would only run on the IBM3090 computers under the JCL operating system”,6 andalthough later versions made it more versatile and user-friendly, it remained for a long time a project developed mainlyin Lund (Sweden), using a set of custom-written tools fordebugging, configuring, updating, and documenting the code.5

In the past few years, particularly since the establishment ofannual MOLCAS developers workshops in 2013, theMOLCAS project has become a much more internationaland collaborative project and has tried to keep up with thetimes and embrace the new tools and technologies that areavailable for code management. Perhaps the most significant ofthe changes, and the one that triggers the need for thispublication, is the choice of an open-source model, asdiscussed in the Introduction.The main codebase of OpenMolcas is written in Fortran

language, something that reflects its history and backgroundand the popularity of Fortran in scientific programming. Newercode is increasingly written in other languages: a number oflibraries and interfaces recently added to OpenMolcas arewritten in C or C++, and the main input parser and driver hasbeen rewritten in Python. The most significant changesintroduced in OpenMolcas with respect to the developmentenvironment are the use of third-party open-source tools forcode management, compilation, and testing. Already in 2013 itwas decided to use git36 as a version control system forMOLCAS; in connection with the release of OpenMolcas, itwas considered important to have a source code repository thatis publicly accessible and not only for developers. As a buildand configuration system, OpenMolcas now uses CMake,37 asystem that also simplifies the combination of differentpackages such as some of the interfaces reported in thispaper. For testing, OpenMolcas uses the verification tools (testsuite and scripts) that were already available in MOLCAS, butit additionally benefits from the continuous integrationfacilities in GitLab,38 the chosen hosting platform, to ensurefast and reliable test runs with minimal maintenance fromOpenMolcas developers.Historically, communication between the different programs

in MOLCAS has been mostly done through files, in text orbinary format. MOLCAS users will be familiar with names like“RunFile”, “InpOrb”, or “JobIph”. These files have theinconvenience that they are specific to MOLCAS and insome cases incompatible between different versions and/orplatforms. With the release of OpenMolcas there is also anincreased effort to make use of more portable and standardizedfile formats for transferring data between programs and for

storing the results of calculations−orbital and CI coefficients,geometries, and energies. In particular, the most importantresults can now be stored in the HDF5 (hierarchical dataformat) format,39 which allows efficient storage and access ofstructured data, including its description. HDF5 files can thenbe accessed in any platform without knowledge of the programthat generated them, since the size and structure of the data arealso part of the format. We consider this an important stepforward toward interoperability with other scientific codes.

3. MULTICONFIGURATIONAL WAVE FUNCTIONMETHODS

Important conceptual bases for understanding the motivationsfor some of the most important features of OpenMolcas areprovided by the definition of configuration and the distinctionbetween static correlation and dynamic correlation. Aconfiguration is a particular way to assign electrons to orbitals,which may be doubly occupied, singly occupied, orunoccupied, and a configuration state function (CSF) is anapproximate wave function with appropriate spin symmetry(singlet, doublet, etc.) and a definite set of orbital occupancies.Static correlation is also called near-degeneracy correlation,left−right correlation, and strong correlation, and it is due tothe interaction of nearly degenerate configurations. A goodzeroth-order wave function for a system with significant staticcorrelation will have appreciable coefficients for two or moreCSFs. Dynamic correlation is the rest of the correlation energy;it includes a variety of correlation effects ranging from veryshort-range (due to the cusp in the correct many-electron wavefunction when two electrons approach one another) to verylong-range (as in dispersion interactions). The slow con-vergence of dynamic correlation with respect to adding moreCSFs is often the bottleneck in making accurate calculations,and the most common way to calculate dynamic correlation isusually to optimize the orbitals with a self-consistent-field(SCF) calculation and then include dynamic correlation byadding CSFs that are excited from a reference SCF wavefunction (single excitations, double excitations, triple excita-tions, etc.); the addition of excited CSFs may be done byperturbation theory, configuration interaction, or coupledcluster40−42 theory. When static correlation is negligible,good results can often be obtained with a single-CSF referencewave function (typically a Hartree−Fock,43−46 HF, wavefunction); methods employing this scheme are called single-reference methods. For strongly correlated systems, thismethod might be inaccurate unless one uses very highexcitation levels (quadruple excitations or higher), which areoften unaffordable. A more efficient way to treat stronglycorrelated systems is to use a multiconfiguration referencewave function; such methods are called multireferencemethods. Systems well treated by single-reference methodswith low levels of excitation are sometimes called single-reference systems, and systems that are much more efficientlytreated with multiconfiguration reference wave functions aresometimes called multireference systems or strongly correlatedsystems. Because many chemical systems of practical interestfeature electronic structures dominated by more than oneelectronic configuration in their ground or excited states, aswell as along reaction pathways in which they are involved, thedevelopment of efficient methods for multiconfiguration wavefunctions and multireference correlation methods is a keychallenge for quantum chemistry; it was one of the primarymotivations for the original development of MOLCAS, and it



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remains a key strength of OpenMolcas. Multiconfigurationalself-consistent field (MCSCF) methods, capable of dealingwith these cases, have been known since the early days ofquantum chemistry, but they gained popularity only when thecomplete active space self-consistent field method (CASSCF)was developed.47−50

Density functional theory presents a similar but conceptuallydistinct scenario. Kohn−Sham density functional theory (KS-DFT)51 does not involve a wave function for the system understudy, but it does involve a Slater determinant that has thesame electron density as the system under study. This Slaterdeterminant may be considered to be a reference wavefunction, and in that sense KS-DFT is a single-referencetheory. Unlike Hartree−Fock theory, though, KS-DFT is inprinciple exact, in that there is an existence theorem for anexact density functional that would yield the correct electrondensity and energy. However, this density functional isunknown and probably unknowable. We must rely onapproximate density functionals (ADFs), and currentlyavailable ADFs are less accurate for strongly correlated systemsthan for single-reference systems.52,53 One way to overcomethis is to develop density functional theories that usemulticonfigurational reference wave functions, and Open-Molcas includes multiconfiguration pair-density functionaltheory (MC-PDFT)54 as an example of this kind of theory.MOLCAS8 (and now OpenMolcas) has always had a strong

focus in multiconfigurational wave function methods and haspioneered the development of cutting-edge methods aiming atcircumventing the exponential scaling limitation in CASSCF.The restricted active space (RAS) SCF,55,56 the generalizedactive space (GAS)57 SCF, and the SplitGAS58 methods areexamples.Multiconfigurational SCF methods, such as the above,

explicitly recover electron correlation within the active spaceand, via the self-consistent orbital relaxation procedure, part ofthe electron correlation outside the active space. In this respectit is to be noticed that the orbital relaxation in MCSCFprocedures is not to be compared to the self-consistent orbitaloptimization in the single-reference HF procedure, as theMCSCF orbital gradient directly depends on the multi-configurational wave function via the generalized Fockoperator. Correlation effects not included at the MCSCFlevel are usually recovered by a posteriori treatments that usethe preceding MCSCF wave function as reference. Examples ofthese methods within the OpenMolcas package are themulticonfiguration pair-density functional theory, MC-PDFT,54,59−61 and second-order perturbation theory methods,such as CASPT2, RASPT2,62−64 and NEVPT2.65−71 Despitetheir broad applicability, these methods rely on the qualitativecorrectness of the reference wave function, or, in the case ofMC-PDFT, its electron density, on-top pair-density, and theratio of these two.72

The rest of this section describes with more detail some ofthe newest approaches implemented in OpenMolcas to workaround the exponential scaling of CASSCF, while keeping itsconceptual simplicity. In particular, the Stochastic-CASSCFmethod (section 3.1) uses a Monte Carlo algorithm to solvethe CI problem in the active space; the density matrixrenormalization group (DMRG)−for which two differentinterfaces are presented (section 3.2)−expresses the wavefunction as a product of matrices, with a reduced number offree parameters with respect to a linear combination ofconfiguration state functions. In addition, the MC-PDFT

capabilities in OpenMolcas (section 3.3) are enhanced withanalytic gradients and state-interaction functionalities.

3.1. The Stochastic-CASSCF Approach. The Stochastic-CASSCF73−76 has been developed since 2015, initially forMOLCAS,8 and is now available in OpenMolcas. The methodretains the simplicity of CASSCF, while circumventing theexponential scaling of the latter. This is obtained by replacingthe Davidson diagonalization technique,77 in its direct-CIimplementation,55,78 with the full-CI quantum Monte Carlo(FCIQMC) algorithm,79−87 while the Super-CI method47−50

is used for the orbital optimization. Starting from the decoupledtwo-step Newton−Raphson method88 for the optimization ofthe orbitals and CI coefficients, the Stochastic-CASSCFprocedure can be summarized in two conceptually simplesteps−(i) the FCIQMC approach to optimize the CI-coefficients and obtain the one- and two-body density matricesand (ii) the Super-CI algorithm for the orbital coefficientsoptimization. The Super-CI step is done according to standardprocedures47,48,50 and will not be described here. TheFCIQMC approach for the optimization of the CI coefficients,instead, will be described below in more detail.The FCIQMC algorithm is a projector technique based on

the imaginary-time Schrodinger equation

τ∂Ψ∂ = − − ΨH E( )S (1)

In this equation the term ES is an energy offset, referred to asthe shif t, that is updated iteratively, converging to the ground-state correlation energy. In the context of FCIQMC, ES playsthe role of population control. Expanding the wave function,Ψ(τ), into a linear combination of Slater determinants, andintegrating eq 1, with a short time step, Δτ, such that the full

propagator τ− −e H E( )S can be approximated to first-order, aniterable working equation is obtained

∑τ τ τ τ τ

τ τ+ Δ = − Δ −

− Δ≠

C C H E C

H C

( ) ( ) ( ) ( )

( )i i ii i

j iij j

S

(2)

A deterministic update of the CI vector from the time τ to(τ+Δτ) would require the storage of several arrays of the sizeof the CI vector and would face the same exponential scalinglimitation of the standard direct-CI approach. Instead, inFCIQMC the concept of walkers is introduced. Walkers aresigned Kronecker delta functions that reside on a specificdeterminant. We define δi,iα as the αth walker residing ondeterminant i . A sign is assigned to each walker, sα = ±1,associated with it. Thus, the signed number of walkers, Ni,which reside on i is given by Ni = ∑αsαδi,iα. The total numberof walkers residing in all relevant determinants is kept to a fixedvalue, Nw = ∑i|Ni|. This value is to be considered anoptimization parameter in addition to the basis set and theactive space. The larger the total number of walkers, thesmaller the error due to the initiator approximation.80 Inpractice, this parameter is gradually increased until con-vergence over the energy estimate is reached.The key point of FCIQMC is to allocate memory only for

the populated determinants, instead of allocating memory forthe entire CI vector. The purpose of FCIQMC is to generatedistributions of walkers such that for a large number of totalwalkers and long imaginary time limit the expectation value ofthe number of walkers on each determinant becomes



E


proportional to the CI coefficient of that determinant, Ci ∝⟨Ni⟩τ. In FCIQMC, walker population dynamics simulates theimaginary-time evolution as given in eq 2. The dynamicsconsists of three steps. In the spawning step child particles aregenerated from their parents into various locations of theHilbert space. In the death step parent walkers arestochastically removed from the simulation. In the annihilationstep parents and newly created walkers residing on the samedeterminant with opposite sign are eliminated.The expectation value of the number of walkers on each

determinant in practical calculations is never explicitlycalculated. Instead derived properties are accumulated oncethe walker population has reached a dynamical equilibrium.One of these properties is the projected energy

=∑

⟨ ⟩τ

τE

N D H D

Ni i i

P0

0 (3)

where the numerator and denominator are accumulated andaveraged separately, D0 is the reference determinant, and ⟨N0⟩τis the time-averaged number of walkers on D0. One- and two-body reduced density matrices (RDMs) are similarlyaccumulated82 to evaluate the CASSCF energy as theexpectation value of the Hamiltonian operator and to computethe matrices required for the orbital optimization step.Electron correlation outside the active space, also known as

dynamic correlation, can be recovered via MC-PDFT,54,59−61

using the Stochastic-CASSCF wave function as reference.3.1.1. Technical Details. The Stochastic-CASSCF method

in OpenMolcas relies on the interface of the NECI program,responsible for the FCIQMC CI algorithm, and the RASSCFprogram of OpenMolcas, responsible for the Super-CI methodas an orbital optimizer. In principle, two installation protocolscan be adopted that are referred to as embedded and uncoupledforms. In the embedded form, the NECI program is treated as asubroutine of the RASSCF program. This form effectively leadsto an automatized version of the Stochastic-CASSCF withinthe OpenMolcas software. In the uncoupled form of Stochastic-CASSCF, NECI is installed as a stand-alone program, and theOpenMolcas−NECI interface is controlled manually by theuser. Dedicated keywords are required in the OpenMolcasinput file to let OpenMolcas produce the relevant files, namelythe file containing one- and two-electron integrals in the MObasis (FCIDUMP) and an input file for the NECI program(see section S1). FCIQMC is then started externally, and thepopulation is grown and stabilized before one- and two-bodydensity matrices are accumulated. Walker population and otherparameters can be tuned at this stage within the FCIQMCcalculation if necessary. Density matrices are then transferredback to OpenMolcas and used to evaluate the Fock matricesnecessary for the Super-CI step. After orbitals have beenrotated, the integral file is updated, and a new FCIQMCdynamics in the active space can be performed. The embeddedOpenMolcas−NECI form is the recommended form for simplecases (small active spaces, small number of walkers, looseconvergence). The uncoupled OpenMolcas−NECI form is therecommended form for difficult applications (large activespaces, large number of walkers, tight convergence).Geometry, spatial symmetry, AO integral evaluation, scalar

relativistic effects, and additional external potentials areevaluated at the OpenMolcas level, and corrections are madeto the FCIDUMP provided to the NECI program.

With the choice of the FCIQMC algorithm as CIeigensolver, the Stochastic-CASSCF method can be appliedto larger active spaces, when compared to the deterministicanalog. Active spaces containing up to 40 electrons and 38orbitals have been reported.75 Integral evaluations and AO−MO transformations can be carried out via resolution-of-identity (RI) Cholesky decomposition (CD) techniques,89−93

available within the OpenMolcas package. The choice of theSuper-CI method for the orbital relaxation together with thedensity fitting techniques enable Stochastic-CASSCF calcu-lations easily with up to 5000 basis functions.

3.1.2. Applications of the Stochastic-CASSCF Method.The Stochastic-CASSCF method has been applied to a numberof chemical problems. Two very recent cases will besummarized in this section, namely the correlation mechanismleading to the large effective antiferromagnetic spin coupling, J,in corner-sharing cuprate solids and the enhanced σ-donation/π-back-donation mechanism that explains the stabilization ofthe intermediate spin-states over the high spin-states in Fe(II)-porphyrin model systems. For a more detailed description ofthese systems and the strategy used within the Stochastic-CASSCF framework we refer to the literature.74−76

Corner-sharing cuprates are of great interest, as they hosthigh-temperature superconductivity upon doping.94 Magneticproperties of undoped cuprates have been rationalized on thebasis of the Heisenberg−Dirac−Van Vleck Hamiltonian withantiferromagnetic nearest-neighbor magnetic coupling con-stant J. The values of J in these systems are among the largestknown, with some variations upon rather small geometricaldifferences.95 From a quantum-chemical standpoint under-standing how J directly depends on electron correlation is achallenging problem that has been investigated by severalresearch groups.96−100 Conventional wave function theorytreatment greatly underestimates the J value. The Stochastic-CASSCF method has been used to investigate this aspect. Acluster containing two CuO4 (or CuO6) units and all adjacentCu2+ and Sr2+ (or La2+) ions has been employed in thesecalculations. The rest of the solid has been modeled by anarray of point charges fitted to reproduce the Madelungpotential due to infinite crystal in the cluster region. It has beendemonstrated that to capture the essential elements of thesuperexchange mechanism it is necessary to explicitly correlatethe 20 valence 3d and double-shell d′ orbitals on the Cu sitestogether with the six 2p and p′ orbitals on the bridging O ion,with a total of 24 electrons, resulting in a CAS(24,26).Calculations with this size of active space would be impossiblewith standard CASSCF technology, the corresponding CI wavefunction containing ∼9 × 1013 Slater determinants. Thecalculation is carried out with ease using the Stochastic-CASSCF method.76

Metal-porphyrins are crucial active sites in many enzymesresponsible for electron transfer, oxygen transport and oxygenreduction. For example, the Fe(II)-porphyrin is the activespecies in cytochrome P450, that binds molecular oxygen andreduces it, leading to P450 Compound 1, responsible for thehydroxylation reaction (insertion of an oxygen atom in a C−Hbond) in alkyl chains. In spite of the numerous experimentaland theoretical data, a definitive understanding of the ground-state electronic structure and its changes along the reactionpathway remain ambiguous, and an electronic explanation forthe relative ordering of the low-lying spin-states remainsunknown. Via large Stochastic-CASSCF(32,34) calculations,74

a complex mechanism has been proposed that demonstrates



F

http://pubs.acs.org/doi/suppl/10.1021/acs.jctc.9b00532/suppl_file/ct9b00532_si_001.pdf


the differential stability of the intermediate triplet spin-stateover the quintet spin-state in a model system of the Fe(II)-porphyrin. This mechanism involves ring correlation at thelevel of the π-system, a correlated breathing process at themetal center and charge-density redistribution. The large(32,34) active space includes the entire π-system and nitrogenlone pairs, the 3d and d′ orbitals at the metal center, and the(4s4p) shell. It has been demonstrated that ring correlationreduces the electron repulsion among the π electrons, makingthe macrocycle a better electron acceptor. The correlating d′orbitals provide the necessary flexibility for the orbitalrelaxation induced by charge-transfer processes from thelone-pairs of the ligand to the metal center. They also accountfor radial correlation, responsible for a lower on-site electronrepulsion at the level of the metal center, and provide a largeroverlap with the π orbitals thus favoring charge-transfer fromthe metal center to the ligand. This complex mechanism can berelated to a correlation enhanced σ-donation/π-back-donationprocess and it is stronger for the triplet spin-state. It wassuggested in ref 74 that smaller active spaces, even uponsecond-order perturbative correction, such as in the conven-tional CASPT2(8,11) are not capable of fully capturing thismechanism (see section 3.2.5 for further discussion). TheStochastic-CASSCF technique has been also used to under-stand the role of semicore electrons. This analysis has beenperformed via a large CAS(40,38) active space.75

3.2. Density Matrix Renormalization Group Inter-faces. After White proposed the DMRG101,102 in 1992,subsequent work103−109 established that DMRG optimizes amatrix product state (MPS). The flexible nature of MPSsallows one to calculate excited states in an efficient manner110

and provides easy access to higher-order reduced densitymatrices and the evaluation of arbitrary correlation functions.The standard definition of an MPS describing a state over Lorbitals or sites is given by

∑ ∑ σψ = ···σ

σ σ σ

−−

M M Ma a

a a a a,...,

1 1L

LL

1 1

11

1 22

1(4)

with basis states σ σσ = ,..., L1 and σ = ↑↓l , ↑ , ↓ , 0 forspatial orbitals. In systems with symmetries, the matrix indicesal are complemented with a quantum number label al →qlαl,

111 leading to a block-sparse structure. Matrix productstates will encode a full CI wave function exactly if the matrixdimension m (the maximum size of the indices al, called“number of renormalized block states” or “bond dimension”) isallowed to grow unrestrictedly. In practice, a finite m is chosenwhich limits the amount of entanglement that the resultingMPS is able to describe between any bipartition of an L-orbitalactive space. For one-dimensional systems, so-called arealaws112,113 prove that the amount of entanglement is constantirrespective of L and that limiting m is therefore a goodapproximation. For the general case in quantum chemistry, theconvergence rate in m depends on molecular topology and thechoice of the orbital basis, influencing the sparsity of the one-and two-electron integrals.114

Analogously to MPS, matrix product operators(MPOs)110,115,116 are defined as

> ∑ ∑ σ σ = ··· ··· ′σσ

σ σ σ σ σ σ

′

′ ′ ′

−− −

W W Wb b

b b b b,...,

1 1L

l ll l

LL L

1 1

11 1

1 1(5)

The efficient determination of the W matrices forHamiltonian operators with long-range interactions such asCoulomb interactions is a nontrivial task.117−119 The DMRGalgorithm formulated explicitly in terms of MPS and MPO issometimes referred to as second-generation DMRG. Com-pared to the original formulation, a pure MPS/MPOimplementation provides additional flexibility by allowingarithmetic operations to be performed on wave functionsand operators independently. These properties are exploited,for instance, to implement a fully relativistic DMRG variant120

in QCMAQUIS (the first such implementation was presentedearlier121), for interfacing with multireference perturbation

Figure 1. Schematic representation of a DMRG-SCF/DMRG-CASPT2 calculation performed by the OpenMolcas−CheMPS2 interface. Only thenumber of renormalized states m must be specified. The user can define optional parameters to speed up the calculation, e.g., orbital ordering,occupation guess, the matrix product state (MPS) checkpoint files from a previous calculation, etc.



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theory122,123 (see section 3.2.3) and for MPS state-interaction124 (see section 3.2.4).This section presents the interface of OpenMolcas with two

established DMRG codes: the CheMPS2 library,125 a first-generation DMRG implementation, based on a MPSrepresentation of wave functions (eq 4), and QCMA-

QUIS,111,117,126 a second-generation program that additionallymakes use of a MPO representation of operators (eq 5). In anutshell, both interfaces can replace the FCI solver fromCASSCF with a DMRG implementation and be used to obtainstate-specific or state-averaged DMRG-SCF wave functionsand perform single-state DMRG-CASPT2 calculations,127

while QCMAQUIS also supports DMRG-NEVPT2, MPSSI,and analytical state-specific gradients for DMRG-SCF, supportfor the recently published analytical state-averaged DMRG-SCF gradients128 will be added in the near future.The CheMPS2 library is a free open-source spin-adapted

implementation of DMRG for ab initio quantum chemistry. Itis designed for high-performance computing, allowing forhybrid MPI and OpenMP parallellization, and can handle ageneral active space up to 40 electrons in 40 orbitals in aDMRG-SCF calculation. In a DMRG-CASPT2 calcula-tion122,127,129,130 requiring the 2- and 3-particle reduceddensity matrices (2- and 3-RDM) as well as the generalizedFock matrix contracted with the 4-RDM (F.4-RDM), an activespace of ∼30 active orbitals can be treated. Figure 1 illustratesa schematic representation of a DMRG-SCF/DMRG-CASPT2calculation performed by the OpenMolcas−CheMPS2 inter-face. DMRG-SCF calculations are performed in a mannersimilar to standard CASSCF calculations except that thenumber of renormalized state m must be provided by the user.Additionally, other parameters controlling the cost andaccuracy of the calculations, e.g., the initial guess of the matrixproduct state, the orbital ordering, the MPS checkpoint files,etc., can be optionally specified. The shared-memory (OMP)CheMPS2 binary is then called and executed. CheMPS2outputs, such as the 2-RDM, are then fed to the RASSCFmodule, allowing the active orbitals to be optimized with thesuper-CI method. If a subsequent DMRG-CASPT2 calculationis required, the active orbitals are transformed intopseudocanonical orbitals in the last DMRG-SCF iteration,and the 3-RDM and F.4-RDM are calculated by CheMPS2based on this pseudocanonical orbital basis. Because thegeneralized Fock matrix is diagonal in the pseudocanonicalorbital basis, the calculation of the full 4-RDM is avoided,which drastically reduces the computational cost.122

Regarding the other DMRG interface described in thispaper, the OpenMolcas−QCMAQUIS interface, it has beendesigned to (i) steer a DMRG calculation with the MPS/MPOprogram QCMAQUIS

111,117,126 and (ii) exchange data such asN-particle reduced density matrices (N-RDMs), fromQCMAQUIS to OpenMolcas. Figure 2 illustrates the capabilitiesof the OpenMolcas−QCMAQUIS interface. DMRG-SCFcalculations with QCMAQUIS are enabled in OpenMolcaswithin the DMRGSCF module by specifying the Active-SpaceOptimizer=QCMaquis keyword which setsQCMAQUIS as default active space solver. Note that theDMRGSCF module shares the same subset of keywords, e.g.,for the wave function model specification, orbital-optimizerconvergence acceleration, and MO analysis, that are availablein OpenMolcas within the complete active space SCFoptimizer module RASSCF. The core of the OpenMolcas−QCMAQUIS interface comprises a subset of Fortran90 modules

which (i) “translate” in an automated fashion the user-providedwave function input data of OpenMolcas into the correspond-ing DMRG input for QCMAQUIS, (ii) generate on the fly theone- and two-electron integrals in the MO basis in thecommonly used format defined by Knowles and Handy,131

(iii) invoke the actual DMRG wave function optimization withQCMAQUIS, and (iv) convert any (user-)requested N-RDM orN′-particle reduced transition density matrix (with N ={1,2,3,4} and N′ = {1,2,3}) from QCMAQUIS to OpenMolcasformat. The complete MPS wave function, optimization input,and output data as well as all molecular properties calculatedby QCMAQUIS are stored in external files in HDF5 format andare therefore available in OpenMolcas for further externalprocessing.

3.2.1. Active Space Selection Based on EntanglementMeasures. The selection of an appropriate active space formulticonfigurational calculations is a task as important as it isdifficult. While several guidelines exist for this selection,132−136

it still requires expertise with trial and error and, hence,constitutes a considerable entry barrier to such calculations.137

An automated protocol has been proposed138 and imple-mented in combination with the OpenMolcas−QCMAQUIS

interface, which selects orbitals for the active space based onorbital entanglement entropies139,140 from a partially con-verged DMRG calculation. This protocol has become possibleowing to (i) the feature of DMRG that allows one to considerlarge active spaces in an exploratory fashion through limitingthe number of sweeps and the block dimension and owing to(ii) the spread of entanglement measures defined for one or apair of orbitals so that rather robust thresholds can be definedthat separate strongly from weakly correlated orbitals, theformer to be chosen for an active space in a final well-converged DMRG or CASSCF calculation.The orbital entanglement entropies are calculated from the

one- and two-orbital reduced density matrices. The elements ofthese matrices can be expressed as expectation values of stringsof creation and annihilation operators and are efficientlycalculated within a DMRG algorithm. The explicit equation forthe single-orbital entropy, on which the selection protocol isbased, reads

∑= −α

α α=

s w w(1) lni i i1

4

, ,(6)

Figure 2. Schematic representation of the capabilities of theOpenMolcas−QCMAQUIS interface. OpenMolcas modules that areinterfaced with QCMAQUIS are listed on the left, and the propertiesthat can be obtained are listed on the right.



H


whereas the two-orbital entropy follows in close analogy as

∑= −α

α α=

s w w(2) lnij ij ij1

16

, ,(7)

In these equations, wα,i and wα,ij are the eigenvalues of the one-and two-orbital reduced density matrices, respectively, i,j areorbital indices, and α is one of the four (or 16) possibleoccupations of a spatial orbital (or a pair of spatial orbitals).Both quantities can be combined to give the mutualinformation which quantifies the entanglement of an orbitalpair

δ= [ + − ] −I s s s12

(1) (1) (2) (1 )ij i j ij ij (8)

Both the mutual information and the single-orbital entropiesare collected in entanglement diagrams such as the one inFigure 3 that provide an intuitive interpretation of the orbitalentanglement.It was demonstrated that CAS wave functions calculated

from these automatically selected active spaces are suitablezeroth-order wave functions for subsequent CASPT2 calcu-lations.141 Furthermore, consistent active spaces for severalelectronic states and along reaction coordinates can beidentified.142 The automated active orbital space selectionhas been implemented as a graphical user interface (GUI)called AUTOCAS143,144 that drives OpenMolcas and QCMA-

QUIS. AUTOCAS is the first computer program that allows one torun multiconfigurational calculations, for which an active spacemust be selected, in a fully automated way. It can be

downloaded separately free of charge from the SCINE Webpage.145

3.2.2. Analytic Ground- and Excited-State DMRG-SCFGradients. A central feature of the traditional multiconfigura-tional wave function toolbox of OpenMolcas is the availabilityof analytic energy gradients (with respect to the nuclearcoordinates) for a single (“state-specific”) or an ensemble(“state-averaged”) of electronic states.146−148 State-specific andstate-averaged gradients are, for instance, key for thecalculation of ground- and excited-state equilibrium structures,reaction paths, and for molecular dynamics simulations.Likewise, excited-state gradients are vital for modeling a widerange of photochemical phenomena, such as photochemicaland photophysical pathways or resonance Raman spectra.Whereas a state-specific DMRG-SCF gradient is ratherstraightforward to implement,149 a formalism for state-averaged DMRG-SCF gradient theory has only been recentlypresented and applied to an optimization of a conicalintersection in the study of the photochemistry of 1,2-dioxetanone.128 The applicability of state-specific DMRG-SCF gradients has been demonstrated for optical spectroscopy,in particular, for resonance Raman spectroscopy,150 and that ofthe state-averaged DMRG-SCF gradients in a conicalintersection optimization.128

In analogy to CASSCF, the calculation of a state-specificDMRG-SCF analytic gradient, available with the Open-Molcas−QCMAQUIS interface, requires the one- and two-particle reduced density matrices for a given target state inaddition to the derivatives of the Hamiltonian, i.e., thederivatives of one- and two-electron integrals, with respect tothe nuclear coordinates.149,150 The latter quantities can becalculated independently of the parent multiconfigurationalwave function. Their calculation is carried out most efficientlyin the AO-basis (to which the reduced density matrices arebacktransformed from the MO basis) which is implemented inthe ALASKA module of the OpenMolcas framework.

3.2.3. Perturbation Theory with DMRG Reference. WhileDMRG-SCF is capable of describing static correlation for largeactive spaces, a significant contribution to electron correlationrequired for quantitative results, namely the dynamiccorrelation, must be accounted for separately. Traditionally,second-order perturbation theory with a zeroth-order Hamil-tonian that describes the static correlation effects has beenmost popular (but coupled-cluster models combined with aDMRG reference have also been considered151−153). Multi-configurational second-order perturbation theory in the formof CASSCF/CASPT2 became the flagship method of the(Open)Molcas quantum chemistry package.154 In the early2000s, Angeli et al.65−68 introduced another formulation of aCAS-based multireference perturbation theory, the (second-order) n-electron valence state perturbation theory(NEVPT2). The main difference of NEVPT2 with respect toCASPT2 is the definition of the zeroth-order Hamiltonian:NEVPT2 uses the Dyall Hamiltonian as a model Hamil-tonian,155 which explicitly considers two-electron interactionsamong the electrons in the active space. The two-electroninteractions in the reference ensure that NEVPT2 is intruder-state free and size consistent.71 NEVPT2 allows for twodifferent contraction schemes to construct the first-order wavefunction: the partially contracted (PC) scheme is identical tothe one in CASPT2, while the strongly contracted (SC) schemeintroduces additional contractions, further simplifying theformalism.

Figure 3. Entanglement diagram obtained from a DMRG wavefunction optimized with the QCMAQUIS program with m = 250 and 10sweeps including all valence orbitals of butadiene that were obtainedfrom a minimal basis Hartree−Fock calculation. A circle with an areaproportional to the single-orbital entropy is associated with eachorbital. The thickness of the connecting lines is proportional to themutual information of an orbital pair. The π-orbitals are those withnumbers 10−13 and would be selected by the automated protocol asimplemented in AUTOCAS.



I


The majority of the DMRG-based perturbation-theorymethods encompass DMRG-CASPT2122,130 (available withboth the OpenMolcas−CheMPS2 and OpenMolcas−QCMA-

QUIS interfaces) and DMRG-NEVPT2123,156 (available with theOpenMolcas−QCMAQUIS interface). One of the major draw-backs of these approaches (or rather of the internal contractionformalism157−159 on which these theories are based) is theirdependence on higher-order N-RDMs (N > 2), whoseevaluation scales as 6 L( )N2 with the number of active orbitalsL. For instance, in the case of the four-particle RDM, itsevaluation scales formally as 6 L( )8 . While for small activespaces (small L) this is not a problem, the fourth-order RDMevaluation becomes quickly the computational bottleneck foractive spaces of more than 20 to 22 orbitals.Apart from the RDM scaling problem, the ability to deal

with large basis sets is crucial for increasing active space sizes.In particular, the transformation of the two-electron integralsfrom the AO to the MO basis becomes a bottleneck. One ofthe flagship features of the OpenMolcas program package isthe Cholesky decomposition of the two-electron integralmatrix, initially proposed by Beebe and Linderberg160 anddeveloped over the years by Aquilante, Lindh, Pedersen, andco-workers for various single- and multireference electronicstructure methods.90,93,161,162 The DMRG-NEVPT2 imple-mentation in QCMAQUIS

123 and the DMRG-CASPT2calculations with the OpenMolcas−CheMPS2 and Open-Molcas−QCMAQUIS interfaces take full advantage of theCholesky decomposition to handle the two-electron inte-grals162 and the DMRG reference wave function (CD-DMRG-NEVPT2, CD-DMRG-CASPT2).3.2.4. State Interaction for Matrix Product States. In the

restricted active space state-interaction (RASSI) approach ofMalmqvist,163 sets of spin-free (SF) eigenstates can be coupledunder the influence of the spin−orbit (SO) coupling, and avariety of highly important molecular properties can beevaluated. A critically important feature of the RASSI programis that the state interaction can be performed among differentsets of SF states with different spin or spatial symmetry, withthe molecular orbitals (MOs) optimized for each set of statesindividually. This implies mutual nonorthogonality of therespective MO bases and requires the transformation of pairsof MO sets to a biorthonormal basis.163 These nonunitarytransformations require a countertransformation of therepresentations of the wave functions such that the physicalcontent of the latter remains intact. This feature was, untilrecently, only available for CI-type wave functions but not forMPSs.The formalism and implementation for MPS-based state

interaction (MPSSI) within QCMAQUIS124 includes the critical

capability to work with nonorthogonal MO sets. The approachdiffers considerably from Malmqvist’s but is likewise based onconstructing biorthogonal MO bases and concomitantcountertransformations of the MPSs. An MPSSI flowchartfor generating SO-coupled wave functions and matrix elementsof operators for molecular properties is shown in Figure 4.Very importantly, the MPSs serve as a complete substitute forthe CI-type states in calculations of SO-coupled wave functionsas well as SF and SO molecular properties with the RASSIprogram. For instance, ref 124 reported relative SF and SOvalence state energies and state compositions for Te2, NpO2

2+,and PuO2

2+, as well as EPR g-factors for the actinyl species,from MPS-SO calculations with QCMAQUIS and the RASSI

program. The MPSSI results were in full agreement withCASSCF and CASPT2.164 Therefore, the full suite of SO-levelmolecular properties and spectra previously available in theRASSI program can now be calculated with the potentially verylarge active spaces that are accessible with MPS wavefunctions.

3.2.5. Applications of DMRG-MRPT2. The case of the spinstates of Fe(II)-porphyrin (FeP) has already been mentionedin the context of Stochastic-CASSCF (section 3.1.2), where itwas proposed (for a smaller model system) that a smallCAS(8,11) active space might be insufficient and a much largeractive space, including all ligand π as well as extra nitrogenorbitals and the 4s4p shell on the metal center, is needed.74

Here we discuss a series of DMRG-SCF and DMRG-CASPT2calculations performed on the lowest triplet (3A2g) and quintet(5A1g) states of FeP with the OpenMolcas−CheMPS2interface. The active space used is CAS(26,27), containing,next to the CAS(8,11) space, 16 P (π,π*) orbitals (all but the2pz of the β carbons of the pyrrole rings). The best DMRGcalculations, using m = 10000 for the CAS(26,27) active space,differ by less than 0.4 kJ/mol from the CAS(8,11) results,giving a quintet−triplet gap of −68.53 kJ/mol with DMRG-SCF and −19.46 kJ/mol with DMRG-CASPT2. Theconclusion is that including the P (π,π*) orbitals on top ofthe CAS(8,11) space has a minor impact on the CASSCF/CASPT2 description of the spin state energetics in FeP. In refs165 and 166 it was shown that an improved description shouldinstead come from more extensive basis sets (extrapolated tothe complete basis set limit) as well as a special (CCSD(T))treatment of the Fe (3s,3p) semicore electrons. All details ofthe calculations, a plot of the active natural orbitals, and adiscussion on the convergence and performance can be foundin the SI.Another example of the OpenMolcas−CheMPS2 interface

shown in the SI is the geometry optimization of ferrocene(Fe(C5H5)2 or FeCp2) at the DMRG-CASPT2 level; thisdemonstrates that the interface can easily be coupled withother modules that require either the 2-RDM or CASPT2 totalenergy. The equilibrium structure of FeCp2 was optimized

Figure 4. Flowchart illustrating the MPS state-interaction (MPSSI)approach to calculate property matrix elements of spin-free and spin−orbit coupled MPS wave functions according to ref 124.



J




(with numerical gradients) employing both standard CASPT2,with a CAS(10,10) active space, and DMRG-CASPT2, with anextended CAS(14,18) active space. The number of renormal-

ized states m is 1000, which gives converged DMRG-CASPT2energies for medium-sized active spaces.129 Further computa-tional details and active natural orbitals can be found in the SI.The key structural parameters of ferrocene, calculated withCASPT2, DMRG-CASPT2, and CCSD(T),167 are shown inTable 1. Interestingly, the results indicate that CASPT2 andDMRG-CASPT2 give almost the same structure. As comparedto CASPT2, DMRG-CASPT2 predicts slightly longer Fe−Cand Fe−Cp bond distances by less than 0.005 Å. The resultsimply that in this simple case, the size of the active space has aminor impact on the optimized geometry. As compared to theCCSD(T) results,167 both CASPT2 and DMRG-CASPT2predict the same d(C−C) and d(C−H), whereas theyunderestimate the Fe−C and Fe−Cp bond distances by∼0.03 Å. All calculations were performed on a single Intel(R)Xeon(R) CPU E5-2680 v3 @ 2.50 GHz machine, 256GBRAM, and 16 MPI processes. A numerical gradient calculationwith the CAS(10,10) active space finishes in around 2 h, whileit takes ∼9 h for the DMRG CAS(14,18) active space. For

comparison purposes, a standard CASPT2 calculation employ-ing the large active space requires a fairly large amount ofmemory (∼20GB per process), consequently, only 12 MPIprocesses can fit the machine, and a numerical gradientcalculation takes almost 8 days with this setting. It is also worthnoting that a geometry optimization cycle using CCSD(T)analytical gradients can take almost 9 days.167

As an example of a DMRG-NEVPT2 calculation with theOpenMolcas−QCMAQUIS interface, we show here results ofCD-DMRG-NEVPT2 for the energy gaps between the lowestquintet, triplet, and singlet states of a large iron(II) complex,FeN2C72H100, shown in Figure 5. This complex was chosen byGuo et al. in reference calculations168 for their DLPNO-NEVPT2 approach169 and by Coughtrie et al. in their paper onembedded multireference coupled cluster theory.170 As thePC-variant of the CD-DMRG-NEVPT2 approach is known tobe unreliable for large active spaces and small bond dimensionsm (due to the incomplete convergence of the zeroth-orderwave function),123 the CD-DMRG-SC-NEVPT2 variant wasapplied, given that it is much less sensitive to the convergenceof the zeroth-order wave function. The active orbitals for theinitial DMRG-SCF step were determined with the automatedactive space selection protocol138 using the AUTOCAS program(see section 3.2.1). The active spaces for all spin states must beequal and, according to the selection protocol, comprise theunion of the active spaces selected for the individual spinstates. In this way, 17 orbitals were selected by the protocol,resulting in an active space of 14 electrons in 17 orbitals.However, upon orbital optimization of the triplet wavefunction, one orbital rotated out of the active space, andhence, the active spaces for different spin states would nolonger be consistent, so the orbital was removed from theactive space for all spin states, resulting in an active space of 14electrons in 16 orbitals. In all calculations, the maximum bonddimension was set to m = 512. Although this is a rather smallvalue for the bond dimension, it was found to be a suitableapproximation in certain cases,123,156 especially if an accuracy

Table 1. Structural Parameters of the EclipsedConformation of Ferrocenea

CASPT2b DMRG-CASPT2c CCSD(T)d

d(Fe−Cp) (Å) 1.610 1.615 1.648d(Fe−C) (Å) 2.016 2.020 2.047d(C−C) (Å) 1.426 1.426 1.427d(C−H) (Å) 1.077 1.076 1.079∠(Cp−H) (deg)e 0.89 1.08 0.52

aIn CASPT2, DMRG-CASPT2, and CCSD(T) calculations, the cc-pwCVTZ basis set (672 basis functions) was used, and all 96electrons were correlated. bStandard CASPT2 calculation withCAS(10,10). cDMRG-CASPT2[m = 1000] calculation with CAS-(14,18). dReference 167. eAngle between hydrogens and cyclo-pentadienyl ring, positive values mean hydrogen is bent toward metal.

Figure 5. Lewis and molecular structure of the FeN2C72H100 complex in ref 169 (Cartesian coordinates were taken from that reference). Colorcode: turquoise−carbon, white−hydrogen.



K



of about 8 kJ/mol is considered to be sufficient. Furthercomputational details are available in the SI.Table 2 contains the quintet−triplet and quintet−singlet

excitation energies calculated with CD-DMRG-SC-NEVPT2and DMRG-SCF along with results from refs 169 and 170obtained with several NEVPT2 and embedded MRCCvariants. The quintet−triplet gap of 165.8 kJ/mol deviatesonly by approximately 10 kJ/mol from the embedded ic-MRCCSD(T) result from ref 170 which may be attributed tothe different basis sets (ANO-RCC-VDZP/VQZP vs def2-TZVP) and the different active orbital spaces from which thereference wave function was constructed. The quality of thereference wave function may be assessed from a comparison ofthe CASSCF and DMRG-SCF excitation energies with thefinal results that include the NEVPT2 contribution: DMRG-SCF excitation energies show a significantly smaller deviationfrom the DMRG-SCF plus NEVPT2 results than theirCASSCF counterparts, which indicates that NEVPT2 benefitsfrom an improved active space for the reference wave functioncalculation and that this can be provided by the suggestedautomated active-orbital selection algorithm. However, thequintet−singlet excitation shows a much larger discrepancybetween the DMRG-NEVPT2 results and those from theliterature. First of all, the CASSCF result for the small activespace deviates even more from the DMRG-SCF result thanwhat was observed for the quintet−triplet excitation.Unfortunately, no NEVPT2 reference data are available forthis higher excitation. Still, the embedded NEVPT2 result fromCoughtrie et al.170 is about 30 kJ/mol larger that theembedded MRCC result, whereas they were very similar inthe singlet−triplet case. This may also be taken as an indicationthat the CAS of the reference was too small because theMRCC approach may compensate for a small-CAS referencewave function through the coupled-cluster excitation hierarchy.Accordingly, the DMRG-NEVPT2 result obtained for thelarger active space is only about 20 kJ/mol smaller than theembedded MRCC result.3.3. Multiconfiguration Pair-Density Functional

Theory. Multiconfiguration pair-density functional theory(MC-PDFT)54,61 combines the advantages of wave functiontheory and density functional theory to provide a method forefficient evaluation of the electronic energy of stronglycorrelated systems. In MC-PDFT the first step is to calculatea multiconfiguration wave function, and in OpenMolcas, thiswave function may be calculated by CASSCF,47 RASSCF,56

GASSCF,57 CAS-CI,55 RAS-CI,55 GAS-CI,171 stochastic-CASSCF73 (section 3.1), and DMRG101,102,109,172,173 (section3.2). CASSCF, RASSCF, and GASSCF are special cases of themulticonfiguration self-consistent-field (MCSCF) method, andone can use either state-specific174 SCF or state-averaged175,176

SCF. The total MC-PDFT energy is expressed as

ρ= + Ψ Ψ + + + [ Π]E V T V V E ,CnnMC MC

ne ot (9)

where the first term corresponds to the nuclear−nuclearinteraction; the second, the third, and the fourth terms,respectively, correspond to the kinetic energy, the nuclear−electron attraction, and the classical Coulomb interaction ofthe electronic charge cloud with itself; and the final term is theon-top energy. In eq 9 the kinetic energy and the classicalelectrostatic energy (nuclear−nuclear repulsion energy,nuclear−electron attraction energy, and classical electron−electron Coulomb energy) are directly obtained from themulticonfigurational wave function, Ψ, while the rest of theelectronic energy is calculated from an on-top densityfunctional that is a functional of the electron density ρ andon-top pair density Π (which represents the probability offinding two electrons on top of each other) of that referencewave function.Computationally, eq 9 reduces to

∑ ∑ ρ ρ= + + + [ Π ∇ ∇Π]E V h D g D D E12

, , ,pq

pq pqpqrs

pqrs pq rsnn ot

(10)

where D is the one-electron density matrix, h and g containrespectively the one- and two-electron integrals, and p, q, r, ands are general orbital indices including inactive and activeorbitals. An MC-PDFT calculation involves two steps: (1)calculation of the multiconfiguration wave function and (2)calculation of the on-top density functional energy. Becausethe postself-consistent-field step is a calculation based on adensity functional, the computational effort for step 2 isgenerally negligible compared to the cost of step 1, and thismakes MC-PDFT computationally more affordable than othermultireference methods such as those based on perturbationtheory or multireference configuration interaction.An MC-PDFT calculation always uses multiconfiguration

wave functions that are spin eigenfunctions. MC-PDFT is freefrom delocalization error,177 which is sometimes considered tobe the most fundamental source of error in KS-DFT withapproximate exchange−correlation functionals. MC-PDFT isalso less prone than KS-DFT to self-interaction error.178

The decomposition of the MC-PDFT energy expression ofeq 9 into its individual components is sometimes important forunderstanding chemical and physical phenomena in molecularsystems.72,179 This decomposition is available in OpenMolcas,along with the tabulation of ρ and Π in Cartesian coordinatesfor plotting and visualization.

3.3.1. On-Top Functionals for MC-PDFT. Due to thesymmetry dilemma, the spin-polarized exchange−correlationfunctionals of KS-DFT are not compatible with the spindensities of a multiconfiguration wave function. To bypass thesymmetry dilemma, one uses translated54 and fully translated60

on-top functionals, which are obtained by translation of KS-DFT exchange−correlation functionals of the spin densitiesand their gradients. Translated functionals are denoted withthe prefix “t”, and fully translated functionals are denoted withthe prefix “ft”. If Exc[ρα(r),ρβ(r),∇ρα(r),∇ρβ(r)] denotes a KS-

Table 2. Excitation Energy in kJ/mol Relative to the Quintet (A5) Ground State of FeN2C72H100a

StDMRG-SC-NEVPT2

this workSC-NEVPT2

ref 169DLPNO-SC-

NEVPT2 ref 169NEVPT2 embCASSCF ref 170

ic-MRCCSD(T) emb PNO-CASPT2 ref 170

CASSCFref 170

DMRG-SCFthis work

A3 165.8 177.2 178.6 178.4 176.6 218.6 146.3A1 241.7 290.3 262.4 346.1 197.8

aResults in refs 169 and 170 were obtained with a CAS(6,5) (and with a def2-TZVP basis set), whereas we chose a CAS(14,16) (and an ANObasis set; see text for further discussion) for the CD-DMRG-SC-NEVPT2 and DMRG-SCF calculations. Abbreviation: “emb”, “embedded in”.



L



DFT exchange−correlation functional, in which ρσ is the spindensity (σ = α,β), and ∇ρσ is the gradient of the density, thetranslated functional is given by

ρ ρ ρ ρ ρ ρ[ |∇ | Π ] = [ ∇ ∇ ]α β α βr r r r r r rE E( ), ( ) , ( ) ( ), ( ), ( ), ( )t t t tot xc (11)

where

ρ ρ ρ ρ ρ ρ

ρ ρ ρ ρ ρ ρ

ρρ

= + Π ∇ = ∇ + Π

= − Π ∇ = ∇ − Π

Π = − Π

α α

β β

ikjjjjj y{zzzzz

r r

r r

f f

f f

f

( )2

(1 ( , )) ( )2

(1 ( , ))

( )2

(1 ( , )) ( )2

(1 ( , ))

( , ) max 0,1 4

t t

t t

2

in which ∇ρ is the gradient of that spin density. For a single-determinant wave function, ≤ρ

Π 142 in all space. However, for a

multiconfiguration wave function, this ratio can be greater thanone. Notice that the translated functionals depend on Π butnot on its gradient and that the left and right derivatives of f arenot equal at =ρ

Π 142 , making the gradient of f discontinuous.

Fully translated functionals also depend on ∇Π, and the firstand the second derivatives of the functional are continuous.Both kinds of functionals have been found to give reasonablygood results for multireference systems.OpenMolcas supports the use of several different on-top

functionals, including a translated local spin-density approx-imation (LSDA), various translated generalized gradientapproximations, and various fully translated generalizedgradient approximations. The following is a list of currentlysupported functionals: tLSDA,54,180,181 ftLSDA,60,180,181

tPBE,54,182 ftPBE,60,182 trevPBE,183,184 ftrevPBE,183,184

tBLYP,54,185,186 ftBLYP,60,185,186 tOPBE,182,187,188 andftOPBE.60,182,187,188

OpenMolcas also supports functionals obtained by scalingthe exchange and/or correlation components of the exchange−correlation functionals prior to translation. The scaling factorsmay be input by the user. This allows the functionals to bemodified into an HLE-type functional,189,190 where thestandard scaling factors are 1.25 for exchange and 0.5 forcorrelation, but other scaling factors may be chosen.3.3.2. Performance of MC-PDFT. MC-PDFT generally

produces results that are comparable in accuracy to completeactive space second-order perturbation theory (CASPT2) butat significantly lower computational cost191 for bondenergies,59,60,177,192,193 spin-multiplet splittings,194−197 andother excitation energies.198−201 These results are summarizedin Table 3. A review containing references for tests andvalidations on electronically excited states up to mid-2018 isavailable in ref 202.3.3.3. Analytical Gradients for SS-CAS-PDFT. OpenMolcas

includes analytical gradients for MC-PDFT when a state-specific CASSCF (SS-CASSCF) wave function is used as areference (SS-CAS-PDFT), and these gradients enable the fastand efficient determination of equilibrium and transition statestructures.205 Because MC-PDFT is a nonvariational method,the computation of the gradient requires the construction of aLagrangian. In general, the computation of analytical gradientscan be performed at least an order of magnitude faster than thecorresponding gradient calculation using a numerical finite-difference method. SS-CAS-PDFT shows good accuracycompared to experimental data for both equilibrium and

transition-state structures. Some illustrative single-processortimings for MC-PDFT calculations employing the tPBE on-topfunctional are shown in Table 4.

3.3.4. Density Matrix Renormalization Group Pair-DensityFunctional Theory (DMRG-PDFT). DMRG (see section 3.2) isa powerful method to treat static correlation, especially inmolecular systems requiring large active spaces. MC-PDFT canbe used in conjunction with DMRG to add additional dynamiccorrelation in an inexpensive way. The first applications206,207

of DMRG-PDFT were based on an interface between theexisting MC-PDFT code in OpenMolcas and the DMRG codein the QCMAQUIS

117,126,208 program. An illustrative example206

of singlet−triplet gaps in polyacenes is shown in Figure 6. ThetPBE/6-31G+(d,p) combination of on-top functional and basisset was used.

3.3.5. State-Interaction Pair-Density Functional Theory.The accurate description of ground- and excited-state PESs incases where two electronic states are strongly interacting (suchas near conical intersections or locally avoided crossings) is achallenge for many electronic structure methods. A newframework, called state-interaction PDFT (SI-PDFT),209 has

Table 3. Mean Unsigned Error (MUE) of MC-PDFT Usingthe tPBE Functional and CASPT2

MUE (eV)

ref system/property PT2 tPBE

54 6 covalent or ionic diatomics/ 0.3 0.3dissociation energy

59 10 transition metals with ligand/ 0.24 0.19average bond energy

194 11 n-acenes (from naphthalene to dodecacene)/ 0.06a 0.13a

singlet−triplet gap193 13 main-group atoms or compounds/ 0.12b 0.24b

singlet−triplet gap196 8 organic rings/ 0.01b 0.10b

singlet−triplet gap199 19 main-group atoms/ 0.22 0.42

first excitation energy201 10 doublet radicals/ 0.14 0.19

first 5 vertical excitation energiesaThe reference wave function is optimized from a GASSCFcalculation.57,203 bThe active space for the reference wave functionuses the correlated-participating-orbital (CPO) scheme.204

Table 4. Illustrative Timings (in s) for a Single GradientComputationa Using a Single Processor

cc-pVDZ cc-pVTZ

activespace symmetry CASPT2 tPBE CASPT2 tPBE

NH3 (6,6) Cs 16 7 32 14HCN (8,8) C2v 9 4 17 10CH2O (12,9) C2v 29 9 29 15HCCH (10,10) D2h 20 6 20 11oxirane (10,10) C2v 55 27 136 45pyrrole (6,5) C2v 87 39 1082 212acrolein (4,4) Cs 558 117 3390 160butadiene (4,4) C2h 85 29 933 137pyridine (6,6) C2v 183 32 2256 156maleicanhydride

(8,7) C2v 123 36 1148 218

aCASPT2: numerical gradient. tPBE: analytical MC-PDFT gradient.



M


been introduced to handle these situations in the case of MC-PDFT. This has been recently implemented in OpenMolcasand successfully applied to several cases, such as thedissociation of LiF209 and the charge-transfer reaction pathof a mixed-valence compound.210 SI-PDFT is an extension ofMC-PDFT that involves the diagonalization of an N × N

effective Hamiltonian to generate a new set of N electronicstates with proper PES topology in regions of strong stateinteraction. This method is similar in objectives andapplication to the “perturb-then-diagonalize” approaches inmultireference perturbation theories such as MS-CASPT2211,212 or MC-QDPT.213,214 An illustrative exampleis shown in Figure 7. The tPBE functional was used for both astandard MC-PDFT calculation and a SI-PDFT calculation.While unphysical curve crossings are present in the standardMC-PDFT treatment, these vanish when the SI-PDFTmethodology is used because the final step in the SI-PDFTmethod is a diagonalization.

4. EXPLORATION OF POTENTIAL ENERGY SURFACESUnderstanding the mechanism, rate, and yield of chemicalreactions requires exploring the potential energy surfaces of

interest. A photochemical reaction involves by definitionseveral PESs; the strong coupling between the differentelectronic states promotes efficient nonradiative electronictransitions.215,216 One distinguishes between internal conversioncaused by nonadiabatic coupling between states of same spinmultiplicity and intersystem crossing caused by spin−orbitcoupling between states of different spin multiplicity.Exploration of PESs can be performed either throughoptimization of specific geometries and paths along thereaction or through simulations of the actual dynamics of thereaction in time.Regarding optimization calculations, OpenMolcas allows

one to optimize minima and transition states, as well asminimum-energy paths and intrinsic reaction paths, using theSLAPAF program. Since recently, it is also possible to optimizeconical intersections thanks to the implementation of thenonadiabatic coupling vector. This new feature is detailed insection 4.1. We mention in passing that the SHARC suite(described in section 4.3) provides a module, alternative to theSLAPAF program in OpenMolcas, that allows the optimizationof excited-state minima, minimum-energy crossing points, andconical intersections, using OpenMolcas for calculating theneeded electronic quantities.Regarding dynamics simulations, direct methods calculate

the PESs as needed along trajectories, sampling only therelevant regions of the PESs. These nuclear trajectories arethen used to describe the nuclear wavepacket motion. Onemajor feature that differentiates the several direct methods ableto describe (nonadiabatic) dynamics is the treatment of thenuclear motion through the basis of nuclear trajectories.217

OpenMolcas allows one to simulate Born−Oppenheimer andsurface-hopping dynamics, by treating the nuclei classically.New features presented in section 4.2 include thermalsampling of the initial conditions, application of externalmechanical forces, and study of isotopic effects. As anelectronic structure package, OpenMolcas is also interfacedto other open-source dynamics codes. The interfaces toSHARC and Quantics are presented in sections 4.3 and 4.4,respectively. SHARC contains a general surface-hoppingalgorithm with arbitrary couplings such as the spin−orbitcoupling, while Quantics allows a quantum treatment of thenuclear motion with the direct dynamics variational multi-configuration Gaussian (DD-vMCG) method.

4.1. Conical Intersection Optimization and Character-ization. A key quantity for describing photochemicalprocesses and near-degeneracy between electronic states isthe nonadiabatic or derivative coupling vector. However, untilnow it was not possible to obtain this quantity with theMOLCAS package, and some approximations had to beemployed for nonadiabatic molecular dynamics8 or for thelocation of conical intersections.218

Since its release, OpenMolcas includes analytical non-adiabatic couplings between state-average CASSCF wavefunctions, as described in ref 219. The implementation isbased on the algorithm used for energy gradients,146,148,220 andit is compatible with both conventional and density-fitted two-electron integrals. The availability of nonadiatabic couplingsallows the use of the projected constrained optimization(PCO) method221,222 for locating minimum-energy conicalintersections (MECI). From the user’s perspective, the onlychange with respect to a single-surface geometry optimizationis the addition of an energy difference constraint−gradients,and nonadiabatic couplings are computed automatically as

Figure 6. Singlet−triplet gaps in polyacenes.

Figure 7. Dissociation of LiF calculated with a (6e,6o) active space.



N


needed. Furthermore, the first-order conical intersectioncharacterization proposed in ref 219 is also reported uponconvergence, allowing the unambiguous description andcomparison of conical intersection topographies. The meaningof the characterization parameters is illustrated in Figure 8.From these, it is possible to determine whether the intersectionis peaked or sloped−in the latter case, it is not a minimum onthe upper surface−and this is available in the output as well.4.2. Dynamics Simulations within OpenMolcas.

Born−Oppenheimer and nonadiabatic dynamics can beperformed within OpenMolcas using the DYNAMIX andSURFACEHOP modules. Transitions between differentelectronic states can be treated by two different approaches:(a) according to the quantification of the temporal rate ofmixing between electronic states223 and (b) following Tully’sfewest switches approach224 where the initial populationdensity matrix is defined at the beginning of the propagation.The modular structure of OpenMolcas allows combining otherfeatures in OpenMolcas with dynamics simulations. Forexample, the ESPF module enables hybrid quantummechanics/molecular mechanics (QM/MM) simulations, orany other electronic structure method available in OpenMolcascan be used to obtain the energy and gradients. Recentdevelopments for molecular dynamics simulations have beenincluded in OpenMolcas. They are summarized below.4.2.1. Initial Thermal Sampling. In order to obtain

statistically significant predictions using molecular dynamicssimulations, an ensemble of trajectories has to be computed(usually hundreds of trajectories, even in simple reactions liketrans−cis photoisomerization225−230). The initial state of thesystem has to be appropriately sampled in order to get a goodrepresentation of nuclear positions and momenta. OpenMolcasallows sampling of the phase-space for a thermally equilibratedstate according to the Boltzmann distribution. Given theminimum energy structure of the system, the absolutetemperature, and a second-order expansion of the potentialenergy as determined with a frequency calculation, initialconditions (i.e., nuclear coordinates and momenta) aregenerated; they can be subsequently used for runningindividual trajectories.4.2.2. Steered Molecular Dynamics. DYNAMIX has the

possibility of including multiple external force pairs, each oneapplied on a specific couple of nuclei. The external forces are

added to the molecular force field defined by the electronicstate. This new feature allows the user to simulate the action ofexternal forces (i.e., mechanical effects) acting over a molecularsystem. The magnitude of each force pair is defined as themagnitude of each force vector acting on each of the twoatoms. Many force pairs can be added, generating a finalexternal force vector by summing up all the components oneach atom. Finally, the applied force pairs can be defined asextension or compression forces. In this way differentmechanochemical setups (e.g., polymer mechanochemistry,sonochemistry, or force probes) can be simulated, in a firstapproach, by adding external forces.Steered molecular dynamics can be performed in ground

and excited states, permitting the study of photochemicalprocesses in strained systems. Photochemistry of mechanicallyaffected systems, i.e., mechanophotochemistry, still remainsquite unexplored, and only a few studies show the interestingproperties of these processes. In this regard, it has been shownthat photochemistry can be effectively modified by mechanicalmeans,231 where the interplay between excited-state dynamicsand external forces is far from being simple and intuitive, as isthe case of a trans to cis photoisomerization of a retinal modelsystem, where an extension force, expected to oppose theformation of the cis isomer, is found to enhance thephotoisomerization, significantly increasing the photoreactionquantum yield.229

4.2.3. Isotope Effect. DYNAMIX is making use of theisotope feature in OpenMolcas which enables the consid-eration of specific isotopes in molecular dynamics simulations.Typically, replacing an atom with an isotope is associated witha kinetic isotope effect which affects the reaction rate of achemical process.

In a recent study of the photoisomerization in rhodopsin,the new feature helped to reveal a new type of kinetic isotopeeffect.232 In this specific case, the hydrogen atoms of the 11-cisisomerizing double bond were replaced by deuterium in threedifferent ways: at both hydrogens and at the hydrogen at eitherthe 11 or 12 position (Figure 9). The molecular dynamicssimulations together with time-resolved measurements showedthat while the reaction rate was mainly unaffected, theisomerization quantum yield was significantly altered by thedeuterium replacements. In fact, remarkably, the quantumyield was found to decrease and increase in the monodeu-terated and bideuterated cases, respectively. This studyprovided further evidence on the importance of the “hydrogen

Figure 8. Illustration of the parameters characterizing a conicalintersection. The red disc represents the horizontal plane E = E×. (a)Energy differences in the x and y directions define the pitch δgh andasymmetry Δgh. (b) The green disc represents the average energyplane, as indicated by the small circles on its edge, and the amountand orientation of its maximum deviation from the red disc define therelative tilt σ and the tilt heading θs.

Figure 9. 11-cis-Retinal protonated Schiff base and the three studiedisotopomers.



O


out of plane” mode for the description of the reactioncoordinate of the photoisomerization of the 11-cis-retinalprotonated Schiff base chromophore of vertebrate visualrhodopsins.233,234

Changing the masses of atoms to “artificial” values may alsobe a useful simulation tool to get insights into the factors thatdetermine the yield of chemical reactions. In a recent study ofthe dissociation of dioxetane compounds, this functionality wasused to understand the effect of methylation.235 Substitution ofhydrogen atoms by methyl groups in dioxetane had beenshown to lead to a longer dissociation time, and it had beenpreviously suggested that this was due to the increase in thenumber of degrees of freedom. To investigate further the effectof methylation, dynamics was simulated for the unmethylated1,2-dioxetane, but where the mass of the four hydrogen atomswas increased to reproduce the moment of inertia of themethyl groups. The simulations suggest that approximately75% of the increase in the dissociation time is actually due to apure mass effect (in contrast to the hypothesis put forward in aprevious theoretical study).4.3. Nonadiabatic Semiclassical Dynamics Interface.

OpenMolcas is interfaced with the ab initio moleculardynamics software suite SHARC (Surface Hopping includingARbitrary Couplings).236−240 Through this interface, non-adiabatic dynamics can be carried out based on the CASSCFand CASPT2 electronic structure methods of OpenMolcas.SHARC is a generalization of the surface hopping

method.224 This allows describing processes such as inter-system crossing or laser-induced transitions, which is notpossible with standard Tully surface hopping. Furthermore, theSHARC software suite can manage large trajectory ensemblesand features many analysis tools.The interface between SHARC and OpenMolcas is based on

file communication between the two programs, see theflowchart in Figure 10. The SHARC dynamics driver (themaster process) writes a request file at each simulation timestep, containing the current geometry and a list of requiredelectronic quantities, such as energies, gradients, spin−orbitcouplings, etc. The interface, a Python script, combines therequest information with the electronic structure settings and

computational parameters. The interface writes the Open-Molcas input files, allowing for computation of multiplegradients or displacements for numerical gradients in a taskparallel fashion. The automatically generated OpenMolcasinput employs the GATEWAY and SEWARD modules forsetup and integrals, the RASSCF module for wave functionsand energies, RASSI for transition dipole moments, spin−orbitcouplings, and wave function overlaps (for nonadiabaticinteractions, see below), and MCLR and ALASKA forgradients. For CASPT2 or MS-CASPT2, the calculation ofnumerical gradients is directly controlled by the SHARC−OpenMolcas interface, allowing a simultaneous calculation ofall gradients of all states and the derivatives of dipole momentsand spin−orbit couplings in a single loop. The nonadiabaticinteraction between the different states is described byemploying wave function overlaps241 from the RASSI moduleor by nonadiabatic coupling vectors from MCLR/ALASKA.QM/MM dynamics (CASSCF/force fields) can also beperformed through the OpenMolcas−Tinker interface. Allcollected electronic structure data is finally communicatedback through the interface to the dynamics driver, whichpropagates the trajectory to the next time step.It is noted that the SHARC−OpenMolcas interface can also

produce Dyson norms to approximately describe ionizationprobabilities (see also section 5.3) by feeding the relevant wavefunction data into the WFOVERLAP program242 within theSHARC suite.

4.4. Quantum Dynamics Interface. While implementinga direct dynamics code for classical nuclei is ratherstraightforward since information is only needed at a givenposition in space, a quantum description of the nuclei has moreproblems. We report here a new interface between Open-Molcas and the Quantics package243 (grown out of theHeidelberg MCTDH package244), which allows a quantummechanical treatment of both electronic and nuclear dynamicsvia the direct dynamics variational multiconfiguration Gaussian(DD-vMCG) method.217,245,246 In practice, Quantics usesOpenMolcas to build a local harmonic approximation of thecoupled PESs, as represented in Figure 11. The dynamics canbe run either in Cartesian coordinates or in normal modes. If

Figure 10. Flowchart of the SHARC−OpenMolcas interface. In step (1), the SHARC dynamics driver sharc.x writes a file with the currentgeometry and requests for quantities (e.g., energies E, spin−orbit couplings (SOC)). This file is read by the interface, together with the templateand resource files that define the level of theory and the computational parameters, respectively. In step (2), all necessary OpenMolcas input filesare written, and step (3) executes the pymolcas driver. The resulting output files are parsed for all requested quantities. Optionally (4), theWFOVERLAP program is executed to obtain Dyson norms. Finally (5), all obtained results are handed back to the SHARC driver.



P


the latter, transformation matrices between the two systems ofcoordinates are set up by reading the output file of a frequencycalculation (using the MCKINLEY module with Open-Molcas).Along the dynamics simulation, Quantics writes input files

from a template input file to run OpenMolcas and compute theadiabatic energies, gradients, Hessians, and couplings for allelectronic states involved. The electronic structure calculationsare performed at the center point of each Gaussian basisfunction, and subroutines in Quantics parse the OpenMolcasoutput files to extract the relevant information.247 Currentlypossible electronic structures methods are HF, CASSCF, andCASPT2. In the latter case, it is only the CASPT2 potentialenergies that are used, while the corresponding energyderivatives and couplings are approximated with their valuesat the CASSCF level.If several electronic states are included in the simulation, the

dynamics is performed in a diabatic basis in order to avoidsingularity issues. A diabatization operation is then realized totransform the adiabatic quantities provided by OpenMolcasinto diabatic ones. Two diabatization schemes are available:the regularization diabatization method,248 which limits theapplication to two electronic states, and the propagationdiabatization method implemented more recently.249 Theformer method requires information about a reference conicalintersection geometry (which can be optimized using theSLAPAF module with OpenMolcas). The latter methodrequires the calculation of the nonadiabatic coupling vector(which can be done using the ALASKA module withOpenMolcas) at every new electronic structure point. Whendealing with states of different spin multiplicity, a diabatization

is performed in each spin-multiplicity subset, and internalconversion within each subset is accounted for as usual insame-spin dynamics. Intersystem crossing is treated bycalculating the spin−orbit coupling terms (which can bedone using the RASSI module in OpenMolcas) at every newelectronic structure point.163

In order to save time and avoid redundant electronicstructure calculations, the electronic quantities calculated byOpenMolcas are stored in a database. New database points areonly calculated if the new nuclear geometry differs by somepredetermined amount from all database geometries (rightbasis function in Figure 11). If it is close enough to somedatabase points (left basis function in Figure 11), the quantitiesare interpolated from the values present in the database.250 It isnoted that the molecular orbital coefficients of the CASSCFcalculations are also stored in the database and used as initialguesses for new electronic structure calculations. Also, insteadof calculating the Hessian at every point, after the first point, anupdate procedure can be performed using the databaseaccording to the Powell method.246,251 By running thedynamics calculation several times, more and more pointsare added to the database. The database “grows” iteratively andreaches convergence once no new database points are needed.

5. SPECTROSCOPIC AND MAGNETIC PROPERTIESThe strong focus of the OpenMolcas package in multi-configurational wave function methods makes it an excellenttool for the computational study of systems where multipleelectronic states play an important role. An obvious case is thestudy of electronic spectra, involving transitions betweendifferent electronic states. A paradigmatic example is the studyof transition metal complexes, where the presence of partiallyfilled d and/or f shells gives rise to many close-lying electronicstates of various spin multiplicities.The workhorse for these investigations, once accurate spin-

adapted wave functions and energies are obtained, is theRASSI program.163 This program can compute the spin−orbitinteraction to yield a set of spin−orbit eigenstates, with the useof atomic mean-field integrals,252 as well as transitionproperties between different wave functions.In this section we report some recent developments and

improvements for the computation of spectroscopic andmagnetic properties in OpenMolcas, from the exact semi-classical treatment of light−matter interaction (section 5.1) toapplications in X-ray spectroscopy (section 5.2), photoelectronspectroscopy (section 5.3), magnetic circular dichroism(section 5.4), molecular magnetism (section 5.5), and in thecontext of ultra-accurate calculations (section 5.6).

5.1. Light−Matter Interaction and beyond the Multi-pole Expansion. All spectroscopies depend on the interactionbetween the system and an external electromagnetic field. Forweak fields that can be treated as perturbations, the oscillatorstrength for a transition from an initial state i to a final state f is

ω δ ω ω∝ −f i U f( ) ( )if if2

(12)

where ωif = (Ef−Ei)/ℏ is the resonance angular frequency, andU is the time-independent part of the electromagneticperturbation. Usually, the field is described by a plane wave,and using only terms linear in field strength U then becomes

∑ ε ε ∝ · · + · × · +ÄÇÅÅÅÅÅÅÅÅ ÉÖÑÑÑÑÑÑÑÑk r p k r k sU

gexp(i )( ) i

2exp(i )( ) c.c.

jj j j j

(13)

Figure 11. Quantics−OpenMolcas interface. At each time step, thedatabase of previously run quantum chemistry calculations is probedto build model coupled potential energy surfaces (PESs). If suitablepoints exist around the desired geometry (Gaussian basis function onthe left-hand side of the figure), a representation of the PES isconstructed using those points. If that region of the coordinate spacehas not been previously explored (right-hand side), an OpenMolcascalculation is performed, and information about this new point isadded to the database.



Q


where the sum goes over all the electrons, k is the wave vector,rj is the position, ε is the polarization vector, pj is themomentum operator, g is the electronic spin g-factor, and sj isthe spin operator, and c.c. is the complex conjugate of theprevious terms.Traditionally, the integrals in eq 12 are evaluated using a

multipole expansion of the electromagnetic wave:

· = + · − · +k r k r k rexp(i ) 1 i( ) 12

( ) ...j j j2

(14)

For the transition moments, taking only the zeroth-orderterm amounts to the electric dipole approximation, while thefirst-order term is associated with electric quadrupole andmagnetic dipole interactions. The multipole expansion leads towell-recognized selection rules that for certain systems andspecific choices of coordinate systems can be directly related toelectronic structure. While the dipole approximation is oftensufficient for the low-energy photons in the optical region,high-energy photons have large k vectors, rapidly oscillatingfields, and higher-order terms in the multipole expansioncannot be ignored. These higher-order terms depend on thechoice of origin, at least in cases where there are nonzero termsof lower order. For weak fields the problem of origindependence was recently solved by Bernadotte et al., byincluding all terms to the same order in the oscillator strengthand not in the transition moments, which for the second orderrequires calculations up to magnetic quadrupoles and electricoctupoles.253 The complete second-order expansion isimplemented in OpenMolcas and has been used to calculateX-ray spectra of several mono- and binuclear iron com-plexes.254,255 Although origin independence was first provenfor the velocity gauge,253 origin independence in a finite basisset can also be accomplished in the length gauge.256 However,what is usually referred to as the length gauge is actually amixed gauge, with the electric components in the length gaugeand magnetic components in the velocity gauge.256 Originindependence, in finite basis sets, is not conserved in thismixed gauge.257 Furthermore, when using the completesecond-order expansion the increased basis set requirementsfor higher-order terms in the multipole expansion must beconsidered.257

A solution to the problems with the multipole expansion isto instead use the plane-wave form of the wave vector directly:the exact semiclassical light−matter interaction258 is originindependent, cannot give negative oscillator strengths, andgives what all terms in the multipole expansion ideally shouldconverge to.258 Further, it shows better numerical stability withrespect to the choice of basis set.259 OpenMolcas now includesan elegant and efficient procedure using a standard Gauß−Hermite quadrature to evaluate the integrals in eq 12 in thisformalism.259 Both electric and magnetic terms are calculatedthis way, with the spin-magnetic term in eq 13 being nonzerowhen the spin−orbit operator in the RASSI module is used.With the exact operator OpenMolcas now calculates the

angular dependence on the transition intensities with respectto both polarization and wave propagation direction. This canbe used to simulate experiments where the system is orientedrelative to the photon beam, e.g., in single crystals or moleculeson surfaces. Although no closed formula for the isotropicallytensor averaged oscillator strengths is known, the exact valuecan be approximated by averaging over different directionsusing a Lebedev grid.259,260 Although the averaging over a gridincreases the computational time compared to the multipole

expansion, this will only rarely affect the timing of the fullelectronic structure calculation. To illustrate the capabilities ofthe exact operator, the angular dependence of the single Cu 1s→ 3d transition in single-crystal [CuCl4]2− has been calculatedand compared to the experimental values, see Figure 12.259

The experimental intensity is distinguished by a 4-foldperiodicity, which was used to assign the electric quadrupolecontributions.261 Further, the orientation of the peak maximacan also be used to identify the symmetry of the singlyoccupied 3d orbital. Calculations with the exact operator showthe same angular dependence, including the 4-fold periodicity,which illustrates how the exact operator includes anyinformation that traditionally has been assigned to multipolecontributions.

5.2. X-ray Spectroscopy and Calculations of ManyCore-Hole Excited States. X-ray spectroscopy involves core-hole states that are at least hundreds of eV higher in energythan the valence excited states. In a complete-active-spaceapproach, these states can be included in the CI expansion byplacing the relevant core orbitals in the active space. As thenumber of excitations from these orbitals can be restricted toone, it is convenient to use a restricted active space (RAS)wave function to simulate X-ray spectra.262−265 For suchprocesses, the usual bottom-up approach that includes all statesis problematic because to reach core-hole excited states allpossible valence excited configurations must be calculated first.A few strategies for generating core-hole states are available, forexample, for some systems where it is possible to use symmetryconsiderations or for very small active spaces.266 However, fora standard-size active space there can be millions of valencestates. Strategies that only reach core-hole states are needed.Two of them recently implemented in OpenMolcas arereported here.A convenient solution to this problem is to define a fixed

orbital function χ and a parameter λ and add a term to themolecular Hamiltonian

λ χ = + [ ]H H Psh (15)

Here, Hsh is the shifted Hamiltonian, and P[χ] is an orthogonalprojection on the space of wave functions where orbital χ isdoubly occupied. This is multiplied by λ and can be

Figure 12. Angular dependence of the 1s → 3d copper K pre-edgetransition in [CuCl4]

2−. Experimental data are normalized K pre-edgepeak heights.261 Calculated values are from the exact semiclassicallight−matter interaction with intensity uniformly scaled to matchexperiment.259 The isotropic contributions are due to vibroniccoupling, which are not included in the calculations.



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implemented by modifying the two-electron MO integrals atthe CI step. These integrals are recomputed at eachmacroiteration while the added term is using the same fixedfunction χ all the time. The penalty term has the effect ofraising the energy of those configurations with a filled core-hole to be above those with a single hole, which are now thelowest in energy. Our experience shows that it is easy to find asuitable shift λ, such that this scheme is stable across changingorbitals and structure, and it has been observed to give only aslight energy variation with the input shift parameter.Obviously, special basis functions may have to be supplied toallow the orbitals of the core-hole states to relax accurately, orif inner-valence correlation is important, e.g., for Auger. Thefollowing example shows the calculation of core−valenceexcitations for H2O close to equilibrium, using a specificallyprepared basis set. Two calculations were performed: onewithout the projected shift and the other with the shift. Theorbital energies in Table 5 show large differences between

without and with the core-hole present. The results of thesubsequent RASSI calculation are displayed in Table 6,showing that it is possible to access states very high in energy(compared with typical valence excitations). For otherexamples, please see the online poster in ref 267.For the general case, another simple technique is to remove

from the CI all configurations with fully occupied core orbitals,the so-called core−valence separation (CVS).268 Because ofthe large energy separation between configurations with andwithout core-holes, the latter do not contribute significantly tothe core-hole excited states. For active-space methods the CVSis closely related to the generalized active space method.57

Using the special case of RAS, the core orbitals can be placedin RAS1 allowing at least singles, for single-core-hole excitedstates, or doubles, for double-core-hole excited states. Thecore-hole excited states are found by applying a core-hole (ch)projection operator Pch

Ψ = ΨPch ch (16)

on the RASSCF calculation, which for single-core-hole excitedstates removes all configurations in Ψ that are fully occupied inRAS1.254,269 This ensures that the lowest-energy state in Ψch isa core-hole excited state in the RASSCF optimization. It isnoted that although the core-hole and valence states arecalculated separately and are thus not orthogonal, transition

intensities can still be calculated in the RASSI module163,270

since the different wave functions use the same CI expansion.In X-ray spectroscopy, even if the computation of all valence

states can be avoided using one of the two strategies discussedabove, the number of excited states needed to describe theprocess of interest can be very large. In transition metalcomplexes, the density of states is very high so that, for an X-ray absorption spectrum that typically spans 10 eV, severalhundred final states are required.266 This creates difficulties forstandard techniques such as a multiroot Davidson77 orDavidson−Olsen algorithm.271 A modified CI algorithm hastherefore been implemented to allow for more efficientcalculations of a large number of states.The Davidson−Olsen algorithm is the quasi-Newton

method applied to the CI problem. It uses an approximateHamiltonian H0, typically diagonal, to compute successivecorrections to an approximate CI vector Cj

(i). The next CIvector Cj

(i+1) is then found by diagonalizing a smallHamiltonian in the basis of all CI vectors at the currentiteration and their corresponding correction vectors. Toimprove convergence, the correction vectors of a few previousiterations are often included. When calculating a very largenumber of states, this scheme (hereafter called Version 1)encounters some problems. First, storing several hundred CI orcorrection vectors from a few iterations can put severeconstraints on memory. Second, the higher-lying statestypically get increasingly difficult to converge, which leads toa large number of CI iterations and makes the CI calculationvery costly. Finally, as the number of previous correctionvectors stored increases, they tend to have significant lineardependence, which can lead to numerical instability during theorthonormalization and sometimes even divergence.The OpenMolcas CI algorithm has been significantly

improved for calculations with hundreds of states. The firstaim was to improve stability of the algorithm to better dealwith linear dependencies in the correction vectors. A simpleand quick fix in order to have a more reliable (but slower)convergence was to reduce the number of past iterations forwhich the CI vectors were saved to two instead of five. Thisversion of the algorithm is called hereafter Version 2. As amore general alternative to this, a more comprehensive changeof the code was realized. While the original implementation(Version 1) saved the last five correction vectors for each state,this new implementation instead sets an upper limit to the totalnumber of correction vectors to around 400, which was foundto be the best compromise between speed, flexibility, andstability. Improvements were also made to reduce the risk ofdivergence in case of near-linear dependency by performingadditional orthogonalizations and normalizations. This leads towhat we call Version 3 of the algorithm. Finally, a veryimportant improvement is to stop computing correctionvectors for converged states. As lower roots convergesignificantly faster than the higher ones, this leads to a drasticreduction of the number of vectors to compute during themajority of the Davidson−Olsen iterations. This final version is

Table 5. Orbitals Used for H2Oa

1a′ 2a′ 3a′ 4a′ 5a′* 6a′* 7a′* 1a″ 2a′′*ground −20.6 −1.36 −0.72 −0.59 0.03 0.03 0.04 −0.51 0.04excited −21.3 −1.78 −1.13 −1.02 −0.07 −0.01 0.01 −0.97 0.01

aAsterisked are valence/Rydberg orbitals, unoccupied in the ground state. Orbital energies (Eh), for the ground state and for the excited states(average, hence approximate).

Table 6. States, Excitation Energies (eV), Dipole andVelocity Einstein A Coefficients (ps−1), and Polarization(Cs)

excitation (RASSI) energy dipole velocity polarization

5a′←1a′ (O 1s) 537.2 0.124 0.139 A′6a′←1a′ (O 1s) 539.1 0.316 0.289 A′7a′←1a′ (O 1s) 539.8 0.065 0.052 A′2a″←1a′ (O 1s) 537.7 0.075 0.065 A″



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referred to as Version 4 and is now the new default option inOpenMolcas.Savings in computational time with the modified CI

algorithm depend on the size of the system and the activespace. The improvement is most significant with a larger activespace because then the CI step dominates the computationalcost. As an example, we simulated the X-ray Kα emissionspectrum of [Fe(bmip)2]

2+, a calculation with ten valence andfour Fe 2p and 1s core orbitals in the active space (145404CSFs for the ground state), using up to 90 core-hole excitedstates in each irreducible representation, which was needed toget the desired orbitals in the active space. The calculationconsists of 1 calculation of the initial 1s core-ionized state and3 independent calculations of the 2p → 1s final states ofdifferent irreducible representations. RASSCF failed to

converge in the old MOLCAS/OpenMolcas implementation(Version 1) because of numerical instabilities. The newalgorithm does not suffer from such issues, and compared tothe Version 2 calculation, the number of CI iterations isreduced by a factor of 3.6, while the saving in computationaltime is a factor of 4, see Figure 13.269 Skipping the calculationof correction vectors for the converged low-lying states reducessignificantly not only the computational time but also thenumber of iterations. The reason is that as the lowest statesconverge, more and more of the vectors used as basis for theDavidson diagonalization correspond to the remaining higherstates, which can thus converge faster.5.3. Photoelectron Spectroscopy and Dyson Orbitals.

For the simulation of molecular ionization processes andphotoelectron spectroscopy, it is routinely necessary toevaluate integrals between wave functions with N and N−1bound electrons. The information about the molecular systemmay then be conveniently compressed into a one-electronquantity

∫ϕ = Ψ Ψ ···−x N x x x x x x dx dx( ) ( , , ..., ) ( , , ..., )fi fN

N iN

N NDO

11

2 3 1 2 2

(17)

commonly known as the Dyson orbital (DO). In the secondquantization formalism, the Dyson orbital ϕf i

DO may in a givenorbital basis ϕm be computed as

∑ϕ ϕ= cfim

fi m mDO

,(18)

= Ψ Ψ−c afi m fN

m iN

,1

(19)

where am annihilates a single electron from orbital ϕm.A new feature now natively available in the RASSI module is

the calculation of multiconfigurational Dyson orbitals includingspin−orbit (SO) coupling, similarly to what has previouslybeen implemented with an external software.272,273 Within theRASSI framework for spin-free (SF) states, eq 19 is efficientlyevaluated within a biorthonormal basis274 ϕA to obtain cf i,ASF . Asthe basis ϕA may vary for different combinations of f and i, theDyson orbitals are subsequently re-expressed as cf i,ZSF in theatomic basis ϕZ through standard transformations. Dysonorbitals for spin−orbit coupled states are obtained assuperpositions of those computed for the spin-free states

∑ ξ ξ= *c cfi Zk l

f k i l kl Z,SO

,, , ,

SF

(20)

where ξf,l denotes expansion coefficient l of spin−orbit state fin the basis of the spin-free states.163 Both types of Dysonorbitals are, optionally, exported in Molden format for furtheruse. The current implementation supports states obtained fromRASSCF and RASPT2 calculations, both with and without theexplicit use of symmetry.Applied to photoelectron spectroscopy, the ionization cross

section σf i of the transition ΨiN → Ψf

N−1 is within the dipoleapproximation proportional to the squared photoelectronmatrix element |Dfi|2. Assuming a simple factorization ψk

elΨfN−1

for the final state and the strong orthogonality condition,275 itcan be written as

σ ψ μ ψ μ ϕ∝ = Ψ Ψ = −D str. orth.k kfi fi f

NiN

fi

2 el 1 el DO 2

(21)

where ψkel is the continuum wave function of the ejected

photoelectron. As ψkel is currently not representable within

OpenMolcas, eq 21 cannot be directly evaluated. However,assuming the transition dipole integral between ψk

el and thenormalized Dyson orbital to be constant, commonly called thesudden approximation, the ionization cross section σf i can beapproximated by the squared norm of the Dyson orbital:

σ ϕ ψ μϕϕ

ϕ∝ | | = || || || ||

≈ || ||

≈´ ≠ÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖ ÆÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖ

D kfi fi fifi

fifi

2 DO 2 elDO

DO

2

constant

DO 2

(22)

The approximate cross sections obtained from eq 22 are inthe RASSI output referred to as Dyson amplitudes and providea direct estimate of the relative cross sections. Note that thesudden approximation is inadvisable in cases where, e.g., thephotoelectron has a low kinetic energy or the Dyson orbitals ofthe relevant transitions have strongly differing characters. Still,Figure 14(a)−(c) illustrates how accurate photoelectronspectra may be obtained even for complex molecules from

Figure 13. Total RASSCF time and total number of CI iterations forthe calculation of X-ray emission spectrum of [Fe(bmip)2]

2+. Version1 refers to the original algorithm. In Version 2, the number of pastiterations for which the correction vectors are saved is reduced.Version 3 contains stability improvements and a cap on the number ofstored vectors instead. Additionally, in Version 4, the corrections arenot computed for the already converged lower-energy states.



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the RASSI Dyson amplitudes. Figure 14(d) shows in additionhow the shape of the Dyson orbitals facilitates spectralassignments. Recent publications have further demonstratedthe applicability of the Dyson orbitals for simulation andanalysis of photoelectron spectra,272,273,276−278 and weexpected them to find their use also for simulations of, forinstance, Auger-electron and nonresonant X-ray emissionspectra.5.4. Magnetic Circular Dichroism Spectroscopy.

Magnetic circular dichroism280 (MCD) spectroscopy probesthe differential absorption coefficient Δε for left versus rightcircular polarized light in the presence of a static magnetic fieldB pointing in the light wave’s propagation direction. It mayreveal more information than regular absorption spectra,especially when spectral bands overlap,281 because the MCDintensities of different transitions may have different signs.MCD is also able to provide information about the magneticproperties of the electronic states. However, MCD spectra canbe complicated to analyze, and the analysis benefits greatlyfrom theoretical support. With commonly applied approx-imations, the MCD for an electronic transition is given by

( ) *ε γΔ = − ∂∂ + +

ÄÇÅÅÅÅÅÅÅÅÅÅ ikjjjj y{zzzz ikjjjjj y{zzzzz

ÉÖÑÑÑÑÑÑÑÑÑÑE

Bf E

E k Tf E

( ) 1 ( )B (23)

Here, γ = 2μBNAπ3 log e/(250hc) in Gaussian units, and f(E) is

a line shape function in terms of the photon energy E.Equation 23 and the expressions for the (, ), and * termswere provided by Buckingham and Stephens.282,283 We focushere on the * term, which dominates the MCD for a moleculewith a degenerate ground state, especially at low temperature.

For a transition from a populated state A with componentsAa and degeneracy gA to state J with components λJ , theisotropic * term in eq 23 reads280

* ∑ λ λ= − ⟨ ′| + | ⟩·[⟨ | | ⟩ × ⟨ | | ′⟩]λ′

L S D Dg

Aa Aa Aa J J Aai3

2A a a, , (24)

Here, L and S are the dimensionless orbital and spin angularmomentum one-electron operators, and D is the one-electronelectric dipole moment operator. The “nonrelativistic withspin” level of theory used for the operators in eq 24 iscommonly applied in calculations of absorption spectra andmagnetic susceptibilities of compounds with even the heaviestelements, and therefore it should be adequate for MCD as well.An orbital angular momentum in the ground state is requiredfor a nonvanishing MCD *-term spectrum. Consequently, ifthe ground state at the nonrelativistic or scalar relativistic (SR)level has only spin degeneracy, the * term is zero. In such acase, MCD * terms may appear via spin−orbit coupling(SOC), if it mixes components of other−orbitally degenerate−spin states into the ground state. Other systems may alreadyhave a ground state orbital angular momentum in thenonrelativistic limit.The functionality of OpenMolcas is very powerful for

investigating MCD spectra theoretically.285 Reference 284reported the first ab initio wave function-based MCDcalculation for an actinide complex, namely for the octahedral5f1 system [UCl6]−, using RASSCF and multistate RASPT2calculations followed by treating the SOC via state interaction.[UCl6]− is one of the aforementioned cases where the * termspectrum is entirely a relativistic effect, i.e., due to SOC. Thecalculated and experimental UV−vis range MCD spectra are

Figure 14. Comparison of photoelectron spectra obtained from Dyson amplitudes as implemented in RASSI and from cited measurements. (a)Valence extreme ultraviolet photoelectron spectrum of H2O (g).272 (b) Fe 2p X-ray photoelectron spectrum of [Fe(H2O)6]

2+.272 (c) I 4d X-rayphotoelectron spectrum of I3−.

279 (d) Dyson orbitals (real part) for selected transitions (53.6 and 56.8 eV) from the I3− calculation shown in panel(c), clearly assigning the features to I 4d ionization from the central (53.6 eV) and terminal (56.8 eV) sites.



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compared in Figure 15. The PT2 spectrum, althoughovercorrecting the blue-shifted SCF spectrum, is in reasonablygood agreement with the experimental MCD (also firstreported in ref 284). A collection of open-source utilitycodes and scripts286 was developed for the purpose ofgenerating MCD intensities conveniently for a large numberof transitions from SO-RASSI calculations. When the keywordPRPR is present in the RASSI input, the matrix elements forthe electric dipole moment, the electron spin, and the orbitalangular momentum for the calculated set of SO states areprinted to a set of files, which is convenient for postprocessing.The MCD intensities for vertical transitions, and broadenedspectra, are then generated with the software tools of ref 286.Code for ( and ) terms287 is also available in the repository.5.5. Molecular Magnetic Properties. In the field of

molecular magnetism, the SINGLE_ANISO module8 hasallowed the computation of the parameters of all magneticHamiltonians and the field- and temperature-dependentthermodynamic properties for mononuclear complexes andfragments. The POLY_ANISO module now allows thecomputation of the magnetic properties of polynuclearcomplexes using a semi-ab initio approach. The necessity forsuch an approach is dictated by the current computationaldifficulties with a full ab initio treatment of polynuclearcompounds containing several transition metals or lanthanideions. In the proposed approach, the first step consists ofdividing the investigated polynuclear compound in severalmononuclear fragments containing one magnetic center only.This fragmentation does not imply that the ligand frameworkneeds to be altered: replacing a neighboring magnetic centerwith its diamagnetic closely related metal ion suffices. Forexample, the effect of a neighboring Er3+ site could bereasonably well simulated by a diamagnetic Lu3+, containing acompletely filled 4f shell. Ab initio calculations using theSINGLE_ANISO module are then performed for each of suchfragments. The features of POLY_ANISO can be divided intothree groups: modeling of magnetic exchange interactions,decomposition of magnetic exchange interactions, andcomputation of magnetic thermodynamic properties. BothSINGLE_ANISO and POLY_ANISO also have the ability to

generate in an automatic way various plots, via an interface tognuplot utility.288

5.5.1. Modeling of Exchange. The total magneticinteraction between two metal centers is a sum of theanisotropic exchange interaction and dipole−dipole magneticinteraction. The dipole−dipole magnetic interaction isevaluated exactly by using the total magnetic dipole momentmatrix elements calculated on the basis of local multiplets ofthe corresponding metal sites available from the fragmentSINGLE_ANISO calculations.The easiest simulation of anisotropic exchange interaction is

achieved within the Lines model,289 involving one effectiveexchange (Lines) parameter per interacting pair. Thisparameter quantifies the hypothetical isotropic exchangeinteraction between the ground-state spins of the correspond-ing magnetic ions, which would arise in the absence of spin−orbit coupling. The anisotropic exchange is modeled via theprojection of this effective isotropic model over chosen spin−orbit multiplets of individual metal centers obtained in thecorresponding fragment SINGLE_ANISO calculations. TheLines model is expected to be accurate in three cases: (i) thetwo interacting sites have uniaxial (Ising) magnetic anisotropy(reduces to generally noncollinear Ising exchange), (ii) the twointeracting sites are completely isotropic (obviously reduces toHeisenberg exchange); and (iii) one site is isotropic while theother possesses Ising magnetic anisotropy (reduces most oftento collinear Ising exchange).290 For all other cases ofintermediate anisotropy of interacting sites, the Lines modelis believed to be a reasonable approximation. The Linesexchange parameters are the only unknown parameters in thisapproach and are usually found from a least-squares fit of theexperimental magnetic data. Alternatively, the Lines exchangeparameters can be extracted/estimated from broken-symmetrydensity functional theory calculations (BS-DFT).291,292 To thisend, the anisotropic magnetic ions are replaced by the closestisotropic metal ions. Then the extracted BS-DFT exchangeparameters J12 for a given pair of ions (1,2) in such ahypothetical complex is converted into the correspondingLines parameter J12L for the original complex with the formulaJ12L = (Si1/Sa1)(Si2/Sa2)J12, where the terms Si and Sa stand forthe spin of the corresponding isotropic ion and the genealogicspin of the anisotropic metal ion in the original compound,respectively. This approach was validated on several com-pounds.293

The POLY_ANISO module contains also several extensionsgoing beyond the simple Lines model: (i) the three-axesexchange Hamiltonian containing three exchange parametersfor each interacting pair, corresponding to the diagonal form(i.e., written in the main exchange axes) of symmetricanisotropic exchange Hamiltonian;294 (ii) antisymmetric orDzyaloshinsky−Morya (DM) exchange interaction, requestingthree parameters (the DM vector); (iii) the general anisotropicnine-parameter exchange model including both previousinteractions; and (iv) biquadratic terms in the exchangeinteraction.294 Another extension of the Lines model concernsmore involved (many-parameter) effective exchange inter-actions in the Lines approach instead of a single-parameterone.The total magnetic interaction matrix is then written

straightforwardly for the basis of the products of the on-siteeigenstates obtained in the fragment ab initio calculations andthen diagonalized.295 The obtained eigenstates alongside withthe spin and magnetic dipole moment integrals expanded in

Figure 15. MCD spectrum (5K) of [UCl6]−. MCD * termcalculations based on RASSCF and RASPT2 calculations of doubletand quartet spin states and treatment of SOC by state interaction withthe RASSI module. Calculated Δε generated from Gaussian-broadened vertical * term spectra. Experiment taken at 7T fieldstrength. Reprinted with permission from ref 284. Copyright 2017PCCP Owner Societies.



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the obtained exchange-coupled basis are then further employedfor the computation of the parameters of all magneticHamiltonians and magnetic properties.5.5.2. Full Decomposition of Magnetic Exchange Inter-

actions Using Irreducible Tensor Technique. The imple-mented models of magnetic interaction (e.g., Lines model)describe the exchange interaction between local pseudo-spins.296,297 The Lines parameters are not directly transferablebetween different compounds (given that their localpseudospins might differ). In order to overcome this problem,the full decomposition of all the interaction modelsimplemented in the POLY_ANISO program has beendeveloped and implemented using the irreducible tensortechnique,298 for all interacting pairs. This results in a set ofparameters J(k1,q1,k2,q2), where ki and qi represent the rankand projection of the local irreducible tensor operator Oki,qi onsite i. The interaction Hamiltonian can be recovered exactly bysumming up all contributions (eq 25). The parameters aregiven using the extended Stevens operator basis296,299,300 andcould be used directly in connection with the EasySpinfunction in MATLAB.301

∑ =H J O Ok q k q

k q k q k q k qexch1,2

, , ,, , , , ,

1 1 2 2

1 1 2 2 1 1 2 2(25)

5.5.3. Thermodynamic Magnetic Properties. The POLY_-ANISO module is able to compute basic static magneticproperties for polynuclear compounds. For the computation ofthe molar magnetization, the Zeeman matrix is built on thebasis of several low-lying energy states (user-defined size) anddiagonalized. The resulting eigenstates are used to computemolar magnetization where Zeeman states are being populatedaccording to the Boltzmann distribution law. Contribution tothe magnetization arising from the states which did not enterexplicitly into the Zeeman matrix is considered within thesecond-order perturbation theory. The computation of molarmagnetization (and torque) takes into account also thecontribution of local excited states which were not accountedfor exchange coupling. Molar magnetic susceptibility (tensorand powder) is computed using the zero-field limit of themolar magnetization, where the implemented expressions donot require the Zeeman Hamiltonian to be explicitlycomputed, enabling significant speedup in computation ofthese properties compared to the computation of molarmagnetization. The magnetic susceptibility is computed inseveral formulations: (i) the derived formulas for zero-fieldlimit, (ii) using the “experimental” formulation as M/B(inaccurate but employed quite often), and (iii) as derivativeof the magnetization with respect to applied field dM/dB (i.e.,the rigorous definition). Intermolecular interactions areaccounted for in a mean-field approach, using a singleparameter (zJ). This parameter is usually quite small inmagnitude and influences magnetic susceptibility, torque, andmagnetization at very low temperatures. For each temperaturepoint, the main values and main axes of the susceptibilitytensor are given.This methodology has been successfully applied for the

investigation of anisotropic magnetism in Dy3 triangles,302 Co7

wheels,303 Ln−(N2)3−−Ln radical-bridged dilanthanides,304

and many other compounds (see Figure 16). POLY_ANISOand SINGLE_ANISO are actively used also for theinvestigation of the structure of blocking barriers of single-

molecular magnets, molecules which display magneticbistability of intramolecular origin.

5.6. Ultrahigh Precision Computer Spectroscopy. Theaccuracy of computer simulations can challenge that ofexperiments, suggesting that it may be worth computing ratherthan measuring some properties. This alternative way toachieve accurate reference data is pursued by some scientistsusing the OpenMolcas platform. In this context we would liketo mention the open-ended implementation of the Open-Molcas platform in handling one-particle electronic basisfunctions with high angular momentum, allowing theevaluation of electron repulsion integrals and their storage inFCIDUMP format, using the same code used for theStochastic-CASSCF approach, discussed in section 3.1. Thisfeature has been used, for example, in the computationalprediction of the carbon atom’s first ionization energy towithin 1 cm−1 accuracy306 from computer spectroscopy.307

This calculation involved six fully correlated electrons in theneutral carbon atom, with an aug-cc-pCV8Z basis set (569basis functions, including up to l angular functions). In thelanguage of CAS spaces, this would be a CAS(6,569), which isby far the largest CAS calculation ever done to microhartreeprecision as far as we are aware.

6. ANALYSIS AND VISUALIZATIONAs discussed above, OpenMolcas offers a broad range ofquantum-chemical methods to describe large molecules withmany correlated electrons and orbitals and to compute manyexcited electronic states efficiently. Setting up these calcu-lations is a nontrivial task, requiring input from the user, whoshould have a sufficient understanding of the system understudy and the method being applied. The analysis can bechallenging if states with different characters mix, if multi-configurational character and double excitations are present, orif a large number of states are to be analyzed.To help the user prepare and analyze calculations,

OpenMolcas contains, or is easily interfaced with, suitabletools. Apart from widespread capabilities, like the computationof Mulliken populations and electric multipole moments,

Figure 16. An example of application of the POLY_ANISO modulefor the ab initio construction of magnetization blocking barrier of aCo(II)−Gd(III)−Co(II) trinuclear complex.305 The magneticrelaxation path, outlining the blocking barrier, is traced by the redlines, whose intensity scales the transition magnetic dipole matrixelements between the connected multiplet states.



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OpenMolcas includes, for example, the LOPROP program308

for computing local properties and supports the generation ofvarious types of orbitals−canonical, natural or localized withseveral methods. The generated orbitals, in the form of lists ofcoefficients, are usually not very informative, and somevisualization tool becomes necessary in order to betterappreciate the shape and structure of the orbitals. Traditionallythis has been done by exporting the orbital and basis-setinformation in Molden309 format. However, it should bepointed out that there are several different ways to choose theorbitals (e.g., state-averaged and state-specific natural orbitalsand natural transition orbitals) and that it is not always clearwhich orbital representation is optimally suited. Therefore, anumber of tools have been developed that allow easilyswitching between these different representations.Recent developments in OpenMolcas provide additional

tools to visualize and manipulate the orbitals (section 6.1), aswell as to analyze their properties (section 6.2), and moreoptions for the analysis of wave functions and transitions,including tools that allow for a completely automatized analysisof the wave function character and provide a rigorous route tocomparing the wave functions produced by different methods(sections 6.3 and 6.4). In this section we describe the mostsignificant of these new capabilities.6.1. Graphical Interface and Orbital Visualization.

OpenMolcas, as other high performance computing programs,requires numeric data as input and can produce a vast quantityof data as output. Raw numeric data (atomic coordinates,orbital coefficients, electron populations, bond orders, etc.) areoften very difficult to comprehend. As a typical example, anessential step in CASSCF and related calculations is selectingthe active space, a task for which an orbital viewer is of greathelp. With MOLCAS, the tools of choice have been eitherthird-party programs capable of computing and renderingisosurfaces “on the fly”, from the basis set and orbitalcoefficients information (in Molden format), or more specificprograms which rely on the precalculation of the volumetricdata and use custom formats.First, in order to make these tasks easier, a program with

GUI: LUSCUS has been developed310 as a stand-aloneprogram (distributed under the Academic Free License(AFL) version 3.0311) convenient for manipulation andvisualization. Although some data produced by quantumchemistry programs can be stored in standardized or well-

established formats, for complex data the formats are oftenunique, and thus GUIs are strongly bound to the computa-tional code. LUSCUS uses a native data format (see Figure 17)to read or write visualizable data, which is supported byOpenMolcas. Other chemical formats can be used withLUSCUS since they are automatically converted to and fromLUSCUS data format by external plug-ins. Plug-ins convertdata on demand without the need of user intervention.Therefore, from the user’s point of view, LUSCUS is capable ofmanipulating a number of chemical formats, making it not onlya graphical front-end for OpenMolcas but also a powerfulgeneral-purpose chemical viewer and editor.Complementarily, a new stand-alone program has been

developed to generate precomputed volumetric data: SAGIT(stand-alone grid independent transformer).312 This code isoperated via command line and can be useful to generatevolumetric data for many orbitals simultaneously or in batchmode, where a GUI is not needed; it can also be used as aplug-in for LUSCUS to visualize orbitals on the fly.With the release of OpenMolcas, a new orbital viewer has

been developed which tries to suit the most typical needs ofOpenMolcas users. Pegamoid313 is written in Python and iseasily installable in modern computer systems. It can openorbital or volumetric files in a variety of formats generated byOpenMolcas. Most significantly, it can open, display, and saveorbital files in the new HDF5 format generated by Open-Molcas, which has several advantages compared to otherformats: (i) as outlined in section 2, it is a portable self-documented format that allows easy transfer and access of data,(ii) it is directly usable by OpenMolcas programs, e.g., asstarting orbitals for the RASSCF program, and (iii) it is notaffected by limitations in the Molden format, like themaximum angular momentum in basis functions or themixture of Cartesian and spherical harmonic type of functions.With respect to formats with volumetric data, the mainadvantage is that its size is not excessive even with large basissets and that the desired quantities to be visualized need not beprecomputed.Apart from simply viewing the orbitals saved in a particular

file, Pegamoid (as well as LUSCUS) can also assign them todifferent orbital spaces (frozen, inactive, RAS1, RAS2, RAS3,secondary, deleted) and save them in a new file that can beused as input for OpenMolcas programs. When the orbital filecontains the necessary information (e.g., an HDF5 filegenerated by a state-average RASSCF run), it is also possibleto visualize, without further postprocessing, natural orbitalsand occupations for the different electronic states, electrondensity differences, natural difference orbitals, and naturaltransition (or binatural) orbitals between the states314 (seeFigure 18).Pegamoid’s goal is to be a quick and convenient tool for

viewing orbitals, selecting active spaces and identifying states.While it has options to control the display quality and features,the generation of high-quality eye-catching pictures oftenrequires the use of other packages that offer a wider range ofoptions. For this purpose, Pegamoid can be used to save thedesired volumetric data in the Gaussian .cube format,21 whichcan then be used by other programs to render pictures withcustomized colors, textures, backgrounds, and additionalelements.

6.2. Property Integrals of Individual Orbitals. Open-Molcas supports the computation of property integrals ofmolecular orbitals in the SEWARD module, including the

Figure 17. Interaction of LUSCUS and various data files: LUSCUScan only operate with .lus files (the native LUSCUS format) andseparate plug-ins (both simple and interactive) are used to convert thedata.



X


multipole moments, Cartesian moments, electric potential,electric field, electric field gradients, diamagnetic shielding,one-electron Darwin contact term, etc. It supports any orbitalscalculated in SCF, RASSCF, CASPT2, or other modules. Withnatural orbitals, the sum of the occupation-number-weightedorbital property integral is the total electronic part of aproperty integral, so the occupation-number-weighted orbitalproperty represents how much each orbital contributes to aproperty integral of an electronic state. For analysis purposes,one has the option of printing, for each molecular orbital, theseproperty integrals weighted by the occupation number of eachorbital or printing these property integrals for each orbitalwithout the weighting. In case one is interested in the propertyintegral for an unoccupied orbital, the nonweighted option isrequired, as otherwise any property integral would trivially bezero. This can be useful when analyzing Rydberg states todetermine which orbitals are Rydberg orbitals. It can also behelpful when deciding which orbitals to include in the activespace of an MCSCF calculation and when determiningwhether the excited states from different MCSCF calculationscorrespond to the same state.315 For these examples, todetermine how diffuse each orbital is, the second Cartesianmoment ⟨r2⟩ and its components (⟨x2⟩, ⟨y2⟩, ⟨z2⟩, ⟨xy⟩, ⟨yz⟩,⟨xz⟩) of each orbital can be calculated. An example input filemaking use of this capability can be found in the SupportingInformation.6.3. Binatural or Natural Transition Orbitals for

Excitation Characterization. There is a frequent need toquantify the difference between two states and to visualize thedifference for easily understood characterization. Assuming wehave at hand correlated electronic wave functions, the usualdifference density is often insufficient. However, recalling thatthe usual natural orbitals are the eigenfunctions of the one-electron reduced density matrix of a single state, a simpleextension is to use the singular value decomposition (SVD) ofthe transition density matrix of two states to characterize theirdifference. This approach was used for single-determinantwave functions316 and for TDDFT and CIS wave functions,317

where the orbitals were named natural transition orbitals, andindependently by Mayer.318 An SVD has also been used torelate orbitals of fragments of a composite molecule.319 Priorusage seems to have been proposed mostly for specificapplications, but in ref 320 it was suggested to use orbitalscomputed as singular vectors of the one-electron transitiondensity matrix as a technical tool to analyze the difference ofpairs of general states, regardless of whether these were single-determinant states or indeed if they were actually computed tobe eigenstates of any Hamiltonian. For general discussion ofthe SVD itself, see, e.g., the articles by Stewart.321,322

The binatural orbitals, or natural transition orbitals, areobtained as the left and right singular functions of the reducedone-electron transition density matrix, γPQ. The binaturalorbitals and their associated amplitudes encode in a compactway the whole γPQ and can be used to analyze the transitionbetween two electronic states.The states would usually be either two noninteracting,

computed states, or they can be two different approximations.They could also be, e.g., states computed with and withoutsome perturbation or indeed any two states that allowed us tocompute the transition density matrix in a suitable basis−maybe but not necessarily obtained as an orbital basis that wasused for the two states. The transition density matrices and thebinatural orbitals are computed by the RASSI program withinOpenMolcas.There is one limitation: what is obtained is the difference of

the states as expressed by a one-electron operator. One caneasily extend this to form two-particle “binatural geminals”.Also, one might have use for “Dyson amplitudes”, as seen insection 5.3, or “one-particle-two-holes” amplitudes, e.g., forcore-hole states.

6.4. Detailed Analysis of Multireference WaveFunctions. The wave function analysis module WFA323

does not only provide visualization methods but also computesa number of quantitative descriptors with the aim ofeliminating personal bias in the assignment of state charactersand allowing for an automated analysis of large data sets. WFAis based on the open-source wave function analysis packagelibwfa.314,324 One focus of libwfa is the analysis of excited statesin terms of two-body electron−hole distributions withinexciton theory.325 To visualize the excitations in real space,binatural or natural transition orbitals317,326 and the morecompact electron and hole densities314 are available. Electron−hole correlation effects can be elucidated using populationanalysis techniques.326,327 Central tools of libwfa are excited-state descriptors quantifying spatial and statistical properties ofexcited states such as exciton size,325 charge-transfer distance,and correlation coefficient.324 Furthermore, it supports theanalysis of one-electron difference density matrices (1DDM)between different states, which is particularly important tostudy orbital relaxation effects.323,328,329 Finally, an effectivenumber of unpaired electrons330,331 can be computed fromnatural orbital occupations. The WFA module is interfaced tothe SCF, RASSCF, and RASSI modules of OpenMolcas.Through the RASSI interface, also other modules can beaccessed such as the CASPT2 module and QCMAQUIS

(DMRG and DMRG-NEVPT2, see section 3.2).Previously, the described analysis methods were used to

analyze single-reference computations addressing a wide rangeof phenomena, such as the analysis of electronic delocalizationin DNA,332 excitonic band structure in conjugated poly-mers,327 and the visualization of solvent effects in push−pull

Figure 18. Pegamoid interface, showing the difference densitybetween two electronic states of uracil (an n → π* transition).



Y

http://pubs.acs.org/doi/suppl/10.1021/acs.jctc.9b00532/suppl_file/ct9b00532_si_002.zip



systems.333 We hope that through the new interface betweenOpenMolcas and libwfa, enabling the use of multireferencemethods, this scope can be significantly extended. Further-more, the implemented analysis methods provide a rigorousroute toward benchmarking excited-state calculations334,335

and can give new insight into specific failures of the differentcomputational protocols employed.329,336

Presenting a practical application of the WFA module, theexcited states of the transition metal complex Re(CO)3(bpy)Cl are analyzed (Figure 19). Computations were performed atthe MS-CASPT2 level using a CAS(12,12) active spaceconsidering 19 singlet and 18 triplet states. The excitationenergies of these states are presented in Figure 19(b),illustrating the high density of low-lying states in this system.A direct analysis of these states in terms of canonical orbitals isnot only tedious but is also quite challenging as the effects ofdifferent interacting configurations have to be disentangled.Therefore, we present here three alternative routes of analyzingthese states, as implemented in the WFA module: a visualanalysis of electron and hole densities, a fragment-baseddecomposition of the excitations, and an analysis in terms ofstatistical descriptors.As a first option, the densities of the excited electron and the

excitation hole, derived from the one-electron transitiondensity matrix (1TDM),314 are computed. These densities,computed for the S1 state of the complex, are shown in Figure19(c). This representation shows the main state character in anintuitive way, i.e., the hole (orange) is located on the Re and Clatoms whereas the electron (cyan) is located on the bipyridyl(bpy) ligand. However, the problem of any visual analysis isthe required time and the dependence of the outcome on thesubjective reasoning of the person analyzing.To overcome these problems of visual analysis, a completely

automatized way to assign the state-character in transitionmetal complexes has been developed,329,335 and it is presentedas a second option here. First, the system needs to be dividedinto different fragments; in the present case a useful partition isthe Re(CO)3 unit (red), the chlorine atom (green), and thebpy ligand (blue), as represented in Figure 19(a). Sub-sequently, the excitation process is partitioned into different

local contributions on the individual fragments and chargetransfer (CT) between them using a 1TDM populationanalysis procedure yielding the so-called charge-transfernumbers.314,326 The outcome of this analysis is presented inFigure 19(d). The left-most bar graph corresponding to the S1state shows that this state is predominantly of Re(CO)3→bpyCT character (yellow bar, 61%) with a secondary contributionof Cl→bpy character (green bar, 26%). More generally, Figure19(d) shows the low-energy states to be generally ofRe(CO)3→bpy character with the exception of T3, which isa locally excited state on bpy (blue bar).A downside of the CT number analysis is that its application

requires an a priori definition of the fragments used for thedecomposition. When intuition is untrustworthy, it is possibleto automatize the fragment definition process.335 Alternatively,we can move away from the population-analysis conceptaltogether leading to the third option, exemplified here throughthe computation of a CT distance in real space. Two differentformulas are employed considering either (i) the distance dh→ebetween the barycenters of hole and electron densities ascomputed from the 1TDM or (ii) the distance dD→A betweenthe barycenters of the detachment and attachment densities337

as computed from the 1DDM.324 The results are shown inFigure 19(e), and without going into too much detail, we wantto mention two observations. First, the CT distance mirrorsCT character as identified in Figure 19(d), i.e., the CTdistances of the local states (red and blue bars) are close tozero while they are significantly enhanced for the CT states.Second, CT distances obtained from the 1TDM and 1DDMdiffer significantly where the latter are always lower. Thisdifference can be interpreted to originate from orbitalrelaxation effects,323,328,329 i.e., the total amount of chargeseparation is lowered as the orbitals adjust to the primaryexcitation process.

7. ADDITIONAL FEATURESIn this final section on new options and utilities inOpenMolcas, we present developments for a new multiscaleoption, accurate and compact basis sets, symmetrization ofwave functions, and simulations of muonic molecular systems.

Figure 19. Analysis of the singlet and triplet excited-states of the Re(CO)3(bpy)Cl complex computed at the MS-CASPT2 level using the WFAmodule: (a) molecular structure and definition of the Re(CO)3 (red), bpy (blue), and Cl (green) fragments; (b) excitation energies; (c) electron(cyan) and hole (orange) densities of the S1 state; (d) decomposition into different local and charge transfer contributions; (e) charge transferdistance using two different descriptors.



Z


7.1. Multiscale Simulations by Frozen-Density Em-bedding Theory. The toolbox of electron correlationmethods present in OpenMolcas would not be completewithout the possibility to combine them in a multiscalefashion. Among the different possibilities for multiscalemodeling in OpenMolcas, the one based on frozen-densityembedding theory (FDET)338−340 provides a robust tool totreat systems and problems of different nature. This multiscaleapproach is a first-principles description of a complex system ascombination of two interacting subsystems. One portion of thesystem, subsystem A, is selected to be described at themolecular orbital level−e.g., through a Kohn−Sham determi-nant,338 or an interacting wave function339−whereas thesecond (subsystem B, so-called “environment”) is accessedthrough a simpler descriptor, namely its electron density. Theinteraction between the electrons of A with subsystem B isnonetheless accounted quantum mechanically through theaction of the following (orbital-free) embedding potential:

∫ρ ρρ

δ ρ ρδρ

δ ρ ρδρ

δ ρδρ

[ ] = +′

| ′ − | ′

+[ ]

+[ ]

+Δ [ ]

r rr

r rr

r r r

v v v

T E F

, , ( ) ( )( )

d

,( )

,( ) ( )

A B B BB

s A B

A

A B

A

A

A

embnuc nuc

nadxcnad

(26)

The first two terms in eq 26 represent electrostaticcontributions due to the nuclei of system B and to its electrondensity ρB(r), respectively. The next two terms are insteadpurely quantum mechanical and originate from the nonadditivecontribution to the noninteracting kinetic energy functionalTs[ρ] and the exchange−correlation functional Exc[ρ],respectively. By nonadditive contribution of a generic densityfunctional G[ρ], we refer to a bifunctional of the electrondensities of A and B, namely to Gnad[ρA,ρB] = G[ρA+ρB] −G[ρA] − G[ρB]. Finally, the last term in eq 26 is not related tothe interaction between the two subsystems, but it ratherensures consistency with the Hohenberg−Kohn theorem uponintroduction of approximations in the description of subsystemA. This term is however neglected in most practicalapplications of the theory.339

The one-electron operator corresponding to the potential ofeq 26 can then be added to the environment-free Hamiltonian(HA) of the quantum mechanical method of choice forsubsystem A in order to proceed with the electronic structurecalculation. Naturally, this way to proceed simplifies thecomputational effort compared to the corresponding quantummechanical calculation on the entire (A + B) system. Onelimitation of the current OpenMolcas implementation is theneed to use explicitly basis functions on B as well as A. For truemultiscale simulations, where for example ρB(r) is obtainedfrom statistical averages,341 the use of a real-spacerepresentation is certainly desirable and will be implementedin a future release of the software. Nonetheless, it should bepointed out that the overhead for the presence of basisfunctions on B is somewhat handled intelligently so thatcorrelation methods need not to include orbitals fromsubsystem B explicitly (see section S15).If the exact functional forms for Ts

nad[ρA,ρB], for Excnad[ρA,ρB],

and for ΔF[ρA] were known, eq 26 would produce an exactquantum mechanical model for the interacting subsystems, butin practice various approximations to these functionals areavailable that guarantee a sufficient accuracy. Some of suchapproximate functionals are included in the current Open-

Molcas implementation of FDET342−345 as detailed in thesoftware documentation. Noticeably, the construction of theembedding potential through eq 26 requires the user to makeanother important choice, namely that of which ρB(r) is to beused. This is a key issue for the correct use of FDET, anddespite the lack of a general answer to the question, the manystudies present in the literature show the general trend that theresults do not depend strongly on the choice of ρB(r).

346 Inpractice, with any physically justified ρB(r), deviations in thepredicted observables are smaller than the errors introduced byother sources, such as the basis set used, the choice of theapproximants for Ts

nad[ρA,ρB] and Excnad[ρA,ρB], or the quantum

mechanical method selected to describe subsystem A.As OpenMolcas is specialized in multiconfigurational wave

function methods,7 the combination with FDET represents asomewhat unique tool for investigating complex systemsespecially in their excited states and for notorious DFT-hardsituations. With this in mind, an effort has been put into thedevelopment of a variant of FDET, known as linearizedFDET,343 that shows some advantages compared to theconventional FDET approach, as it inherits useful properties ofthe corresponding wave function method. Linearizationconsists of approximating the nonadditive functionals so thatthey become linear in ρA(r), by means of a Taylor expansionaround a reference density ρA

ref(r), which can be assumed notto differ significantly from the stationary density of theembedded subsystem. Among the many advantages oflinearized FDET, we point out the fact that the eigenstatesof the embedded Hamiltonian are orthogonal, hence they canbe directly related to the true ground and excited states of theembedded subsystem. Also, differences in the computedeigenenergies of the embedded Hamiltonian can be used toassign excitation energies in much the same way as it is donefor the calculation on the isolated system A.

7.2. Development of New ANO Basis Sets. Basis setsbased on the atomic natural orbitals (ANO) approach347 arethe workhorse of the MOLCAS/OpenMolcas program pack-ages. The ANO contraction scheme offers a fast convergencetoward the uncontracted form of the basis set. Thus, ANObasis sets even in small contracted forms−say valence double-ζplus polarization functions (VDZP)−usually are able to yieldaccurate results, which makes them the perfect choice forcalculations where the bottleneck lies in the correlation part.With the focus on fast convergence by contraction andapplication in accurate correlated calculations, ANO basis setswere allowed to contain a larger number of primitive basisfunctions than other comparable basis sets.348,349

Accompanying the nonrelativistic ANO-L basis set,350 themore compact ANO-S basis set351 had been available, thusallowing more economic treatment of large molecular systemswhere the integral computation time can rival the correlationpart. For relativisitic calculations, however, so far only therelativistic core-correlated ANO-RCC basis set352 had beendeveloped for the second-order Douglas−Kroll−Hess (DKH2)Hamiltonian. The large size of this basis set in terms ofprimitive basis functions made integral calculations forextended systems rather costly. To make also relativisticcalculations for large systems more economic, a new compactrelativistic basis set, the ANO-XS basis set,353 has recently beendeveloped. The number of primitive basis functions in theANO-XS basis set is about half that of the ANO-RCC basis set.This greatly reduces the computational costs in the integral



AA



computation time, especially when used in combination withCholesky decomposition.The ANO-XS basis set is available for light atoms (Z ≤ 20).

Its main intent is to be used in combination with the largerANO-RCC basis set in relativistic calculations containingheavy atoms such as the chromium complex shown in Figure20, constituting a part of a metal−organic framework (MOF).When using the ANO-RCC basis set with VDZP contractionfor all atoms, the total number of primitive basis functions is4573. Replacing the basis set for all atoms except chromiumwith the ANO-XS basis set reduces the number to 2014.7.3. Symmetrization of Wave Functions. At the early

stage of quantum chemistry, the symmetry of molecules playedan important role as an efficient way to reduce the size of thecomputational problems. Today this aspect is not as important,since the absolute majority of computed systems has nosymmetry elements except of identity.Many computational codes, including OpenMolcas, use only

Abelian point groups (D2h and subgroups) for calculations. In

this case the result of any symmetry operation on a matrixelement is a multiplication by 1 or −1.If a molecule has a symmetry higher than D2h, the

computational code will use a lower symmetry, and as aresult, the computed density or wave function can present abroken symmetry. In some cases, this deviation fromsymmetrical solution will be small, but in other cases thiserror will be significant.Rewriting the computational codes for the purpose of

handling an arbitrary point group is a rather complicatedprogramming task. Instead, there is a more simple approach,which involves symmetrization of the computed wave functionaccording to any specified symmetry, e.g., the symmetry of themolecular structure. The procedure of symmetrization of thewave function can be applied at the end of the calculation orduring the iterative procedure of computing the wave function(e.g., in a self-consistent cycle). The details of the algorithm ofsymmetrization are described in ref 354. Prior to an SCFcalculation the point group is determined. The orbital spacesfor each irreducible representation (irrep) are then calculatedfor the supplied basis. At each iteration the symmetric orbitalsare projected out and partner functions determined. Co-efficients are then averaged over the partner functions. Sincethe dimension of each irrep space is known, the symmetriza-tion step will also verify that the projected subspaces belong tothe correct symmetry species. This is required since norestriction is placed on the rotations during the iterations, andlarge symmetry breaking may occur.A mathematical library (libMSYM) has been designed for

two independent tasks: automatic detection of the point groupand symmetrization of the computed wave function accordingto a given symmetry.355 Version 2.0 of the library can beintegrated into the OpenMolcas code and can produce SCFcalculations with wave function symmetrized at any SCF cycle.The usage of libMSYM is especially important if approximateintegrals are used in the calculations. Two examples belowdemonstrate the abilities of the library.As a first example, in tetrafluoromethane (CF4), belonging

to the Td point group, the HOMO orbitals belong to the T1irrep. After removing three electrons from the molecule and

Figure 20. Basis set combination of ANO-RCC and ANO-XS for thecalculation of an MOF template.

Figure 21. Orbitals for CF4 (point group Td) before (upper row) and after (lower row) symmetrization.



AB


running a UHF/DFT calculation the symmetry will break eventhough T1 is 3-dimensional. By applying symmetry restrictionsin the SCF calculation the charges will be evenly distributedbetween 3 orbitals of T1 symmetry as can be seen in Figure 21.The second example is a symmetry-broken solution. The

structure of the (NO2)2 dimer is totally symmetric and belongsto the D2h point group. However, if we add a positive charge,the wave function (at the UHF level) becomes unstable andthe electron is localized at one of the monomers. By enforcingthe symmetry with the libMSYM library, we obtain a fullysymmetric solution. The total dipole moment is of the order of10−13 D.7.4. Calculations of Muonic Atoms and Molecules.

The overwhelming majority of chemical studies, experimentaland computational, are concerned with “normal” matter,constituted of electrons, protons, and neutrons. However, alarge number of other elementary and composite particles areknown which could, in principle, replace the ordinaryFermions, although their lifetimes are usually so short thatthe practical significance is doubtful. An outstanding case isthat of the muon, whose lifetime of 2.2 μs has allowed itsdetection combined with normal matter. The muon is a lepton,with similar properties to the electron, but with a mass about200 times larger. Muons can replace electrons in normalmolecules, and due to their heavier mass, their density isconcentrated much closer to the nuclei. The radiation emittedas a muon falls down to lower energy levels can giveinformation about the nuclear charge distribution and, tosome extent, about the electronic structure. Recent develop-ments in muonic X-ray spectroscopy for nondestructiveelemental analysis356−358 have spurred the interest oftheoreticians to predict the properties of muonic atoms andmolecules.Calculations of systems including muons and electrons are

now possible with OpenMolcas.359 There are a few issues totake care of. One-particle and exchange integrals are alwaysvanishing between electrons and muons, since they aredistinguishable, but Coulomb integrals are always nonzero,regardless of which particles they refer to. Due to the heavymass of the muons, the Born−Oppenheimer approximationbecomes less valid, especially for the lighter nuclei, whichcannot be assumed infinitely heavier than the muons. Toaccount for this, OpenMolcas allows including the so-calledfinite nuclear mass correction (FNMC)360 term to the kineticenergy. In addition, since the muonic density is much moreconcentrated close to the nucleus, the muonic energy levels aremuch more sensitive to the nuclear size and chargedistribution, and a pointlike nuclear model is expected to beinaccurate. In OpenMolcas finite nuclear size effects can beincluded through a Gaussian charge distribution. Finally, thedescription of muons requires the use of specialized basis sets,with exponents much larger than those typically used forelectronic basis functions. Such a basis set has been recentlydeveloped for elements from H to Ar and Cu.359 Initialcalculations for atoms and molecules where one electron wasreplaced with one muon showed satisfactory agreement withexperimentally available data, suggested a mislabeling of one ofthe peaks for Cu and allowed the estimation of the sensitivityof muonic X-ray energies to chemical environment.

8. SUMMARYOpenMolcas−an open-source community development plat-form−is based on the source code of the MOLCAS package.

The program suite is now also a fully operational user’salternative. This move has been done under the leadership ofthe (Open)Molcas developers to promote the use and thedevelopment of the multiconfigurational wave functionparadigm. The report includes technical details of thedevelopment platform and also a large number of novelimplementations. For the first part it was described how theOpenMolcas code is developed and managed. This arrange-ment will allow easier contributions from new developers and amore direct interaction with other software, as alreadyexemplified by some of the interfaces reported in this work.For the second part it is reported an array of new options inOpenMolcas, which include new techniques in wave functionmodels, tools for molecular dynamics and potential energysurface exploration, methods for computing spectroscopic andmagnetic properties, utilities for graphical representation ofresults, and other additional new features. These items aresummarized below.The range of applicability of the CASSCF procedure has

been extended by a number of new implementations andinterfaces. The report describes in some detail the Stochastic-CASSCF approach, two DMRG options−the OpenMolcas−CheMPS2 and the OpenMolcas−QCMAQUIS interfaces−andnew features available for the MC-PDFT method. Further-more, new tools for exploration of the electronic PES andvarious types of MD have been implemented. Morespecifically, we report on availability of analytical nonadiabaticcoupling vectors and associated optimization and character-ization of conical intersections, on new options for theDYNAMIX module, and on two interfaces for moleculardynamics−SHARC for semiclassical dynamics and Quanticsfor on-the-fly quantum dynamics. A number of options andfeatures have been implemented to improve the capacity ofOpenMolcas to address issues associated with different typesof spectroscopy. These options include the use of the exactsemiclassical operator for the light−matter interaction, thetreatment of core-holes, the generation of Dyson orbitals, thesimulations of MCD spectra, the POLY_ANISO module forthe simulation of magnetic properties of polynuclearcomplexes, and the use of OpenMolcas in association withultra-accurate simulations. The OpenMolcas package hasadditionally been supplemented with a list of new tools forpostcalculation analysis. These include LUSCUS−a generalpurpose GUI, a stand-alone grid generator SAGIT, the orbitalvisualizer Pegamoid, the generation of natural transitionorbitals in the RASSI module, and the WFA module toanalyze excited-state simulations. Finally, some additionalfeatures were described, such as the ANO-XS basis set, amultiscale option using frozen-density embedding techniques,implementation of higher point groups through the libMSYMlibrary, and options to handle both the negatively chargedmuons and electrons at the same time.To conclude, we have reported on the new development and

application environment of the open-source quantum chemicalelectron structure and molecular dynamics simulation packageOpenMolcas. This report described a robust software develop-ment environment. This and the documentation of a largearray of new developments should stimulate more developersto join the OpenMolcas community, in particular thoseinterested in multiconfigurational electron structure theoryand various types of molecular dynamics and spectroscopies.Moreover, to the reader, and potential user, we hope that thecase has been made that OpenMolcas−a free-of-charge tool−



AC


can provide a strong and dynamic platform for state-of-the-artsimulations. The user can virtually get access to new optionsand features as soon as the developer deems them stable foruse in production calculations. Looking toward the future andnew development one would in particular look forward to arobust implementation of a QM/MM interface, a betterplatform infrastructure to facilitate efficient massively parallelsimulations, integral-direct RI techniques, and the use of newtechnologies, such as machine learning, to accelerate currentprocedures but also open for new ways in which ab initio dataare used.

■ ASSOCIATED CONTENT*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acs.jctc.9b00532.

Example inputs for Stochastic-CASSCF, DMRG-CASPT2, DMRG-SCF, MPSSI, DMRG-NEVPT2,MC-PDFT, conical intersections, SHARC interface,light−matter interaction, Dyson orbitals, orbital proper-ties, binatural orbitals, WFA, and FDET; computationaldetails for DMRG-MRPT2 examples discussed in text;and discussion on convergence and performance forFe(II)-porphyrin calculation with OpenMolcas−CheMPS2 (PDF)Input files (ZIP)

■ AUTHOR INFORMATIONCorresponding Author*E-mail: [email protected] Fdez. Galvan: 0000-0002-0684-7689Morgane Vacher: 0000-0001-9418-6579Jochen Autschbach: 0000-0001-9392-877XJie J. Bao: 0000-0003-0197-3405Sergey I. Bokarev: 0000-0003-0779-5013Rebecca K. Carlson: 0000-0002-1710-5816Sijia S. Dong: 0000-0001-8182-6522Andreas Dreuw: 0000-0002-5862-5113Luis Manuel Frutos: 0000-0003-1036-7108Laura Gagliardi: 0000-0001-5227-1396Angelo Giussani: 0000-0002-9452-7641Leticia Gonzalez: 0000-0001-5112-794XStefan Knecht: 0000-0001-9818-2372Giovanni Li Manni: 0000-0002-3666-3880Marcus Lundberg: 0000-0002-1312-1202Philipp Marquetand: 0000-0002-8711-1533Stefanie A. Mewes: 0000-0002-6538-9876Jesper Norell: 0000-0003-1058-2588Massimo Olivucci: 0000-0002-8247-209XMarkus Oppel: 0000-0001-6363-2310Felix Plasser: 0000-0003-0751-148XMarkus Reiher: 0000-0002-9508-1565Andrew M. Sand: 0000-0002-7166-2066Igor Schapiro: 0000-0001-8536-6869Prachi Sharma: 0000-0002-1819-542XDonald G. Truhlar: 0000-0002-7742-7294Liviu Ungur: 0000-0001-5015-4225Alessio Valentini: 0000-0002-0882-9417

Tomasz A. Wesołowski: 0000-0001-5792-4616Alexander Zech: 0000-0002-4332-8546J. Patrick Zobel: 0000-0002-9601-7573Roland Lindh: 0000-0001-7567-8295Present Addresses9Theoretical Division, Los Alamos National Laboratory, LosAlamos, NM 87545, USA.14Materials Science Division and Center for MolecularEngineering, Argonne National Laboratory, Lemont, IL60439, USA.21Division of Chemical Physics, Chemical Center, LundUniversity, SE-221 00 Lund, Sweden.22Department of Chemistry, University of Washington, Seattle,WA 98195, USA.31Department of Chemistry, Butler University, 4600 SunsetAvenue, Indianapolis, IN 46208, USA.33Chemical Sciences Division, Lawrence Berkeley NationalLaboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA.34Theoretical Chemistry and Biology, School of EngineeringSciences in Chemistry, Biotechnology and Health (CBH),Royal Institute of Technology (KTH), SE-106 91 Stockholm,Sweden.35Lehrstuhl fur Theoretische Chemie, Ruhr-UniversitatBochum, D-44780 Bochum, Germany.FundingK.P. and Q.M.P. thank the Flemish Science Foundation(FWO) and KU Leuven Postdoctoral mandates (PDM/16/112) for funding. The computational resources and servicesused in DMRG-CASPT2 calculations were provided by theVSC (Flemish Supercomputer Center), funded by theHercules Foundation and the Flemish Government-depart-ment EWI. The Zurich group was supported by theSchweizerischer Nationalfonds (Proj. no. 200021_182400).L.F. acknowledges the Austrian Science Fund FWF for aSchrodinger fellowship (Project no. J 3935). J.A. acknowledgessupport from the U.S. Department of Energy, Office of BasicEnergy Sciences, Heavy Element Chemistry program, undergrant DE-SC0001136. The work at the University ofMinnesota was supported by the Air Force Office of ScientificResearch under grant no. FA9550-16-1-0134 and by theNational Science Foundation under grant no. CHE-1746186.M.G.D., L.K.S., M.V., M.L., M.G., E.K., and R.L. acknowledgefinancial support from the Stiftelsen Olle Engqvist Byggmas-tare, the Knut and Alice Wallenberg Foundation (grant no.KAW-2013.0020), and the Swedish Research Council (2012-3924). The project R-143-000-A65-133 of the NationalUniversity of Singapore is gratefully acknowledged. Thecomputational work for the ANISO section was fullyperformed on resources of the National SupercomputingCentre, Singapore (https://www.nscc.sg). R.L. and I.F.G.acknowledge funding from the Swedish Research Council(grant 2016-03398). L.Go., S.M., P.M., and M.Op. thank theUniversity of Vienna for generous support and all the pastSHARC developers. A.G. is thankful for funding from theEuropean Union’s Horizon 2020 research and innovationprogramme under the Marie Sklodowska-Curie Grant Agree-ment no. 658173. The Spanish MINECO grant CTQ2016-80600-P is acknowledged. The ERC Starting Grant “Photo-Mutant” is acknowledged. S.M., F.P., and L.Go. gratefullyacknowledge funding from the Austrian Science Fund (FWF)within project I2883. M.Ol. is grateful for grants NSF CHE-CLP-1710191, NIH GM126627 01, a USIAS 2015 fellowship,



AD

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http://pubs.acs.org/doi/abs/10.1021/acs.jctc.9b00532



mailto:[email protected]

http://orcid.org/0000-0002-0684-7689

http://orcid.org/0000-0001-9418-6579

http://orcid.org/0000-0001-9392-877X

http://orcid.org/0000-0003-0197-3405

http://orcid.org/0000-0003-0779-5013

http://orcid.org/0000-0002-1710-5816

http://orcid.org/0000-0001-8182-6522

http://orcid.org/0000-0002-5862-5113

http://orcid.org/0000-0003-1036-7108

http://orcid.org/0000-0001-5227-1396

http://orcid.org/0000-0002-9452-7641

http://orcid.org/0000-0001-5112-794X

http://orcid.org/0000-0001-9818-2372

http://orcid.org/0000-0002-3666-3880

http://orcid.org/0000-0002-1312-1202

http://orcid.org/0000-0002-8711-1533

http://orcid.org/0000-0002-6538-9876

http://orcid.org/0000-0003-1058-2588

http://orcid.org/0000-0002-8247-209X

http://orcid.org/0000-0001-6363-2310

http://orcid.org/0000-0003-0751-148X

http://orcid.org/0000-0002-9508-1565

http://orcid.org/0000-0002-7166-2066

http://orcid.org/0000-0001-8536-6869

http://orcid.org/0000-0002-1819-542X

http://orcid.org/0000-0002-7742-7294

http://orcid.org/0000-0001-5015-4225

http://orcid.org/0000-0002-0882-9417

http://orcid.org/0000-0001-5792-4616

http://orcid.org/0000-0002-4332-8546

http://orcid.org/0000-0002-9601-7573

http://orcid.org/0000-0001-7567-8295

https://www.nscc.sg


and a MIUR Department of Excellence grant. S.I.B. acknowl-edges financial support from the Deutsche Forschungsgemein-schaft, grant No. BO 4915/1-1. J.N. acknowledges financialsupport from the Swedish Research Council (contract VR2015-03956) and the Helmholtz Virtual Institute VI419“Dynamic Pathways in Multidimensional Landscapes”. A.Z.,F.A., and T.A.W. acknowledge support from Fonds NationalSuisse (FNS) under grant 200020-172532.NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSM.G.D., L.K.S., M.V., M.L., M.G., E.K., and R.L. thank TrondSaue for useful discussions.

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AF

https://t1.chem.umn.edu/hondoplus

https://t1.chem.umn.edu/hondoplus

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https://www.mrcc.hu

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http://www.petachem.com/products.html

http://www.petachem.com/products.html

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https://cmake.org

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https://www.hdfgroup.org/


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https://scine.ethz.ch/static/download/qcmaquis_manual.pdf

https://scine.ethz.ch/static/download/qcmaquis_manual.pdf


(214) Nakano, H. MCSCF reference quasidegenerate perturbationtheory with Epstein−Nesbet partitioning. Chem. Phys. Lett. 1993, 207,372−378.(215) Malhado, J. P.; Bearpark, M. J.; Hynes, J. T. Non-adiabaticdynamics close to conical intersections and the surface hoppingperspective. Front. Chem. (Lausanne, Switz.) 2014, 2, DOI: 10.3389/fchem.2014.00097.(216) Jasper, A. W.; Kendrick, B. K.; Mead, C. A.; Truhlar, D. G.Non-Born−Oppenheimer Chemistry: Potential Surfaces, Couplings,and Dynamics. In Modern Trends in Chemical Reaction Dynamics;World Scientific: 2004; pp 329−391, DOI: 10.1142/9789812565426_0008.(217) Vacher, M.; Bearpark, M. J.; Robb, M. A. Direct methods fornon-adiabatic dynamics: connecting the single-set variational multi-configuration Gaussian (vMCG) and Ehrenfest perspectives. Theor.Chem. Acc. 2016, 135, 187.(218) Maeda, S.; Ohno, K.; Morokuma, K. Updated BranchingPlane for Finding Conical Intersections without Coupling DerivativeVectors. J. Chem. Theory Comput. 2010, 6, 1538−1545.(219) Fdez. Galvan, I.; Delcey, M. G.; Pedersen, T. B.; Aquilante, F.;Lindh, R. Analytical State-Average Complete-Active-Space Self-Consistent Field Nonadiabatic Coupling Vectors: Implementationwith Density-Fitted Two-Electron Integrals and Application toConical Intersections. J. Chem. Theory Comput. 2016, 12, 3636−3653.(220) Lischka, H.; Dallos, M.; Szalay, P. G.; Yarkony, D. R.; Shepard,R. Analytic evaluation of nonadiabatic coupling terms at the MR-CIlevel. I. Formalism. J. Chem. Phys. 2004, 120, 7322−7329.(221) Anglada, J. M.; Bofill, J. M. A reduced-restricted-quasi-Newton−Raphson method for locating and optimizing energycrossing points between two potential energy surfaces. J. Comput.Chem. 1997, 18, 992−1003.(222) De Vico, L.; Olivucci, M.; Lindh, R. New General Tools forConstrained Geometry Optimizations. J. Chem. Theory Comput. 2005,1, 1029−1037.(223) Vreven, T.; Bernardi, F.; Garavelli, M.; Olivucci, M.; Robb, M.A.; Schlegel, H. B. Ab Initio Photoisomerization Dynamics of a SimpleRetinal Chromophore Model. J. Am. Chem. Soc. 1997, 119, 12687−12688.(224) Tully, J. C. Molecular dynamics with electronic transitions. J.Chem. Phys. 1990, 93, 1061−1071.(225) García-Iriepa, C.; Marazzi, M.; Zapata, F.; Valentini, A.;Sampedro, D.; Frutos, L. M. Chiral Hydrogen Bond EnvironmentProviding Unidirectional Rotation in Photoactive Molecular Motors.J. Phys. Chem. Lett. 2013, 4, 1389−1396.(226) Gozem, S.; Melaccio, F.; Valentini, A.; Filatov, M.; Huix-Rotllant, M.; Ferre, N.; Frutos, L. M.; Angeli, C.; Krylov, A. I.;Granovsky, A. A.; Lindh, R.; Olivucci, M. Shape of Multireference,Equation-of-Motion Coupled-Cluster, and Density Functional TheoryPotential Energy Surfaces at a Conical Intersection. J. Chem. TheoryComput. 2014, 10, 3074−3084.(227) Marchand, G.; Eng, J.; Schapiro, I.; Valentini, A.; Frutos, L.M.; Pieri, E.; Olivucci, M.; Leonard, J.; Gindensperger, E.Directionality of Double-Bond Photoisomerization Dynamics In-duced by a Single Stereogenic Center. J. Phys. Chem. Lett. 2015, 6,599−604.(228) Manathunga, M.; Yang, X.; Luk, H. L.; Gozem, S.; Frutos, L.M.; Valentini, A.; Ferre, N.; Olivucci, M. Probing the Photodynamicsof Rhodopsins with Reduced Retinal Chromophores. J. Chem. TheoryComput. 2016, 12, 839−850.(229) Valentini, A.; Rivero, D.; Zapata, F.; García-Iriepa, C.;Marazzi, M.; Palmeiro, R.; Fdez. Galvan, I.; Sampedro, D.; Olivucci,M.; Frutos, L. M. Optomechanical Control of Quantum Yield inTrans−Cis Ultrafast Photoisomerization of a Retinal ChromophoreModel. Angew. Chem., Int. Ed. 2017, 56, 3842−3846.(230) Gueye, M.; Manathunga, M.; Agathangelou, D.; Orozco, Y.;Paolino, M.; Fusi, S.; Haacke, S.; Olivucci, M.; Leonard, J.Engineering the vibrational coherence of vision into a syntheticmolecular device. Nat. Commun. 2018, 9, 313.

(231) García-Iriepa, C.; Sampedro, D.; Mendicuti, F.; Leonard, J.;Frutos, L. M. Photoreactivity Control Mediated by Molecular ForceProbes in Stilbene. J. Phys. Chem. Lett. 2019, 10, 1063−1067.(232) Schnedermann, C.; Yang, X.; Liebel, M.; Spillane, K. M.;Lugtenburg, J.; Fernandez, I.; Valentini, A.; Schapiro, I.; Olivucci, M.;Kukura, P.; Mathies, R. A. Evidence for a vibrational phase-dependentisotope effect on the photochemistry of vision. Nat. Chem. 2018, 10,449−455.(233) Schapiro, I.; Ryazantsev, M. N.; Frutos, L. M.; Ferre, N.;Lindh, R.; Olivucci, M. The Ultrafast Photoisomerizations ofRhodopsin and Bathorhodopsin Are Modulated by Bond LengthAlternation and HOOP Driven Electronic Effects. J. Am. Chem. Soc.2011, 133, 3354−3364.(234) Gozem, S.; Luk, H. L.; Schapiro, I.; Olivucci, M. Theory andSimulation of the Ultrafast Double-Bond Isomerization of BiologicalChromophores. Chem. Rev. (Washington, DC, U. S.) 2017, 117,13502−13565.(235) Vacher, M.; Farahani, P.; Valentini, A.; Frutos, L. M.;Karlsson, H. O.; Fdez. Galvan, I.; Lindh, R. How Do Methyl GroupsEnhance the Triplet Chemiexcitation Yield of Dioxetane? J. Phys.Chem. Lett. 2017, 8, 3790−3794.(236) Richter, M.; Marquetand, P.; Gonzalez-Vazquez, J.; Sola, I.;Gonzalez, L. SHARC: Ab Initio Molecular Dynamics with SurfaceHopping in the Adiabatic Representation Including ArbitraryCouplings. J. Chem. Theory Comput. 2011, 7, 1253−1258.(237) Mai, S.; Marquetand, P.; Gonzalez, L. A general method todescribe intersystem crossing dynamics in trajectory surface hopping.Int. J. Quantum Chem. 2015, 115, 1215−1231.(238) Mai, S.; Marquetand, P.; Gonzalez, L. Nonadiabatic dynamics:The SHARC approach. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2018,8, No. e1370.(239) Mai, S.; Richter, M.; Heindl, M.; Menger, M. F. S. J.; Atkins,A.; Ruckenbauer, M.; Plasser, F.; Oppel, M.; Marquetand, P.;Gonzalez, L. SHARC2.0: Surface Hopping Including Arbitrary Couplings− A Program Package for Non-Adiabatic Dynamics; 2018. https://www.sharc-md.org (accessed Sept 19, 2019).(240) SHARC GitHub repository. https://github.com/sharc-md/sharc (accessed Sept 19, 2019).(241) Granucci, G.; Persico, M.; Toniolo, A. Direct semiclassicalsimulation of photochemical processes with semiempirical wavefunctions. J. Chem. Phys. 2001, 114, 10608−10615.(242) Plasser, F.; Ruckenbauer, M.; Mai, S.; Oppel, M.; Marquetand,P.; Gonzalez, L. Efficient and Flexible Computation of Many-ElectronWave Function Overlaps. J. Chem. Theory Comput. 2016, 12, 1207−1219.(243) Worth, G. A.; Giri, K.; Richings, G. W.; Burghardt, I.; Beck, M.H.; Jackle, A.; Meyer, H.-D. QUANTICS Package, Version 1.1;University of Birmingham: Birmingham, U.K., 2015. http://chemb125.chem.ucl.ac.uk/quantics/ (accessed Sept 19, 2019).(244) Worth, G. A.; Beck, M. H.; Jackle, A.; Meyer, H.-D. MCTDHPackage, Version 8.3; 2003. http://www.pci.uni-heildelberg.de/tc/usr/mctdh/ (accessed Sept 19, 2019).(245) Burghardt, I.; Meyer, H.-D.; Cederbaum, L. S. Approaches tothe approximate treatment of complex molecular systems by themulticonfiguration time-dependent Hartree method. J. Chem. Phys.1999, 111, 2927−2939.(246) Richings, G. W.; Polyak, I.; Spinlove, K. E.; Worth, G. A.;Burghardt, I.; Lasorne, B. Quantum dynamics simulations usingGaussian wavepackets: the vMCG method. Int. Rev. Phys. Chem. 2015,34, 269−308.(247) Although not implemented yet, one could read the relevantinformation instead from the HDF5 standard format files generatedby OpenMolcas.(248) Koppel, H.; Schubert, B. The concept of regularized diabaticstates for a general conical intersection. Mol. Phys. 2006, 104, 1069−1079.(249) Richings, G. W.; Worth, G. A. A Practical DiabatisationScheme for Use with the Direct-Dynamics Variational Multi-



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http://dx.doi.org/10.3389/fchem.2014.00097

http://dx.doi.org/10.3389/fchem.2014.00097

http://dx.doi.org/10.1142/9789812565426_0008

http://dx.doi.org/10.1142/9789812565426_0008

https://www.sharc-md.org

https://www.sharc-md.org

https://github.com/sharc-md/sharc

https://github.com/sharc-md/sharc

http://chemb125.chem.ucl.ac.uk/quantics/

http://chemb125.chem.ucl.ac.uk/quantics/

http://www.pci.uni-heildelberg.de/tc/usr/mctdh/

http://www.pci.uni-heildelberg.de/tc/usr/mctdh/


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2017; 2017. http://www.molcas.org/VV/poster2_Gothenburg.pdf(accessed Sept 19, 2019).(268) Cederbaum, L. S.; Domcke, W.; Schirmer, J. Many-bodytheory of core holes. Phys. Rev. A: At., Mol., Opt. Phys. 1980, 22, 206−222.(269) Delcey, M. G.; Sørensen, L. K.; Vacher, M.; Couto, R. C.;Lundberg, M. Efficient calculations of a large number of highly excitedstates for multiconfigurational wavefunctions. J. Comput. Chem. 2019,40, 1789−1799.(270) Malmqvist, P.-Å.; Roos, B. O. The CASSCF state interactionmethod. Chem. Phys. Lett. 1989, 155, 189−194.(271) Olsen, J.; Jørgensen, P.; Simons, J. Passing the one-billionlimit in full configuration-interaction (FCI) calculations. Chem. Phys.Lett. 1990, 169, 463−472.(272) Grell, G.; Bokarev, S. I.; Winter, B.; Seidel, R.; Aziz, E. F.;Aziz, S. G.; Kuhn, O. Multi-reference approach to the calculation ofphotoelectron spectra including spin−orbit coupling. J. Chem. Phys.2015, 143, 074104.(273) Grell, G.; Bokarev, S. I.; Winter, B.; Seidel, R.; Aziz, E. F.;Aziz, S. G.; Kuhn, O. Erratum: ”Multi-reference approach to thecalculation of photoelectron spectra including spin-orbit coupling” [J.Chem. Phys. 143, 074104 (2015)]. J. Chem. Phys. 2016, 145, 089901.(274) Malmqvist, P. Å. Calculation of transition density matrices bynonunitary orbital transformations. Int. J. Quantum Chem. 1986, 30,479−494.(275) Cremaschi, P. A theoretical analysis of the four-photonionization spectrum of the nitric oxide molecule. J. Chem. Phys. 1981,75, 3944−3953.(276) Moguilevski, A.; Wilke, M.; Grell, G.; Bokarev, S. I.; Aziz, S.G.; Engel, N.; Raheem, A. A.; Kuhn, O.; Kiyan, I. Y.; Aziz, E. F.Ultrafast Spin Crossover in [FeII(bpy)3]

2+: Revealing Two CompetingMechanisms by Extreme Ultraviolet Photoemission Spectroscopy.ChemPhysChem 2017, 18, 465−469.(277) Golnak, R.; Bokarev, S. I.; Seidel, R.; Xiao, J.; Grell, G.; Atak,K.; Unger, I.; Thurmer, S.; Aziz, S. G.; Kuhn, O.; Winter, B.; Aziz, E.F. Joint Analysis of Radiative and Non-Radiative Electronic RelaxationUpon X-ray Irradiation of Transition Metal Aqueous Solutions. Sci.Rep. 2016, 6, 24659.(278) Norell, J.; Grell, G.; Kuhn, O.; Odelius, M.; Bokarev, S. I.Photoelectron shake-ups as a probe of molecular symmetry: 4d XPSanalysis of I3− in solution. Phys. Chem. Chem. Phys. 2018, 20, 19916−19921.(279) Josefsson, I.; Eriksson, S. K.; Ottosson, N.; Ohrwall, G.;Siegbahn, H.; Hagfeldt, A.; Rensmo, H.; Bjorneholm, O.; Odelius, M.Collective hydrogen-bond dynamics dictates the electronic structureof aqueous I3−. Phys. Chem. Chem. Phys. 2013, 15, 20189.(280) Piepho, S. B.; Schatz, P. N. Group theory in spectroscopy; JohnWiley & Sons: New York, 1983.(281) Neese, F.; Solomon, E. I. MCD C-Term Signs, SaturationBehavior, and Determination of Band Polarizations in RandomlyOriented Systems with Spin S ≥ 1/2. Applications to S = 1/2 and S =5/2. Inorg. Chem. 1999, 38, 1847−1865.(282) Buckingham, A. D.; Stephens, P. J. Magnetic Optical Activity.Annu. Rev. Phys. Chem. 1966, 17, 399−432.(283) Stephens, P. J. Theory of Magnetic Circular Dichroism. J.Chem. Phys. 1970, 52, 3489.(284) Gendron, F.; Fleischauer, V. E.; Duignan, T. J.; Scott, B. L.;Loble, M. W.; Cary, S. K.; Kozimor, S. A.; Bolvin, H.; Neidig, M. L.;Autschbach, J. Magnetic circular dichroism of UCl6

− in the ligand-to-metal charge-transfer spectral region. Phys. Chem. Chem. Phys. 2017,19, 17300−17313.(285) Bolvin, H. Theoretical Determination of the Excited Statesand of g-Factors of the Creutz−Taube Ion, [(NH3)5-Ru-pyrazine-Ru-(NH3)5]

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http://www.molcas.org/VV/poster2_Gothenburg.pdf

https://github.com/jautschbach/mcd-molcas.git


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III4 Heptanuclear Wheel: Lack of SMM Behavior

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http://gnuplot.sourceforge.net

http://gnuplot.sourceforge.net

http://dx.doi.org/10.1007/978-3-642-74599-7

http://dx.doi.org/10.1007/430_2014_171

http://dx.doi.org/10.1002/9781118571767.ch6

http://dx.doi.org/10.1142/0270

http://dx.doi.org/10.15278/isms.2017.MI08

https://sourceforge.net/projects/sagit/

https://pypi.org/project/Pegamoid


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